Integrand size = 22, antiderivative size = 267 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=-\frac {2 a d e^2 \sqrt {e x}}{b^2}+\frac {2 c e (e x)^{3/2}}{3 b}+\frac {2 d (e x)^{5/2}}{5 b}+\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{9/4}}-\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{9/4}}+\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} b^{9/4}} \] Output:
-2*a*d*e^2*(e*x)^(1/2)/b^2+2/3*c*e*(e*x)^(3/2)/b+2/5*d*(e*x)^(5/2)/b+1/2*a ^(3/4)*(b^(1/2)*c-a^(1/2)*d)*e^(5/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/ a^(1/4)/e^(1/2))*2^(1/2)/b^(9/4)-1/2*a^(3/4)*(b^(1/2)*c-a^(1/2)*d)*e^(5/2) *arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(9/4)+1/2 *a^(3/4)*(b^(1/2)*c+a^(1/2)*d)*e^(5/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e* x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/b^(9/4)
Time = 0.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.67 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=\frac {(e x)^{5/2} \left (4 \sqrt [4]{b} \sqrt {x} (-15 a d+b x (5 c+3 d x))-15 \sqrt {2} a^{3/4} \left (-\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{30 b^{9/4} x^{5/2}} \] Input:
Integrate[((e*x)^(5/2)*(c + d*x))/(a + b*x^2),x]
Output:
((e*x)^(5/2)*(4*b^(1/4)*Sqrt[x]*(-15*a*d + b*x*(5*c + 3*d*x)) - 15*Sqrt[2] *a^(3/4)*(-(Sqrt[b]*c) + Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]* a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*a^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*Arc Tanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(30*b^(9/4 )*x^(5/2))
Time = 1.05 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.32, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {552, 27, 552, 27, 552, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {2 e \int \frac {5 (e x)^{3/2} (a d-b c x)}{2 \left (b x^2+a\right )}dx}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \int \frac {(e x)^{3/2} (a d-b c x)}{b x^2+a}dx}{b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (-\frac {2 e \int -\frac {3 a b \sqrt {e x} (c+d x)}{2 \left (b x^2+a\right )}dx}{3 b}-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \int \frac {\sqrt {e x} (c+d x)}{b x^2+a}dx-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \int \frac {a d-b c x}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {e \int \frac {a d-b c x}{\sqrt {e x} \left (b x^2+a\right )}dx}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \int \frac {a d e-b c e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 d (e x)^{5/2}}{5 b}-\frac {e \left (a e \left (\frac {2 d \sqrt {e x}}{b}-\frac {2 e \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{b}\right )-\frac {2}{3} c (e x)^{3/2}\right )}{b}\) |
Input:
Int[((e*x)^(5/2)*(c + d*x))/(a + b*x^2),x]
Output:
(2*d*(e*x)^(5/2))/(5*b) - (e*((-2*c*(e*x)^(3/2))/3 + a*e*((2*d*Sqrt[e*x])/ b - (2*e*(-1/2*(Sqrt[b]*(c - (Sqrt[a]*d)/Sqrt[b])*(-(ArcTan[1 - (Sqrt[2]*b ^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4) *b^(1/4)*Sqrt[e]))) + (Sqrt[b]*(c + (Sqrt[a]*d)/Sqrt[b])*(-1/2*Log[Sqrt[a] *e + Sqrt[b]*e*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2]*a^( 1/4)*b^(1/4)*Sqrt[e]) + Log[Sqrt[a]*e + Sqrt[b]*e*x + Sqrt[2]*a^(1/4)*b^(1 /4)*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/2))/b)))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Time = 1.70 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-\frac {2 \left (-3 b d \,x^{2}-5 c b x +15 a d \right ) x \,e^{3}}{15 b^{2} \sqrt {e x}}-\frac {a \left (-\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right ) e^{3}}{b^{2}}\) | \(318\) |
| derivativedivides | \(-\frac {2 \left (-\frac {b d \left (e x \right )^{\frac {5}{2}}}{5}-\frac {b c e \left (e x \right )^{\frac {3}{2}}}{3}+a d \,e^{2} \sqrt {e x}\right )}{b^{2}}+\frac {2 a \,e^{3} \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\) | \(323\) |
| default | \(-\frac {2 \left (-\frac {b d \left (e x \right )^{\frac {5}{2}}}{5}-\frac {b c e \left (e x \right )^{\frac {3}{2}}}{3}+a d \,e^{2} \sqrt {e x}\right )}{b^{2}}+\frac {2 a \,e^{3} \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\) | \(323\) |
| pseudoelliptic | \(\frac {e^{2} \left (-\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) c \sqrt {2}\, a e}{2}+\frac {\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) d \sqrt {\frac {a \,e^{2}}{b}}\, \sqrt {2}\, a}{2}+a \sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+a \sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \sqrt {e x}\, \left (a d -\frac {x b \left (\frac {3 d x}{5}+c \right )}{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{2}}\) | \(328\) |
Input:
int((e*x)^(5/2)*(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/15*(-3*b*d*x^2-5*b*c*x+15*a*d)*x/b^2/(e*x)^(1/2)*e^3-a/b^2*(-1/4*d/e*(a *e^2/b)^(1/4)*2^(1/2)*(ln((e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/ b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arc tan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4 )*(e*x)^(1/2)-1))+1/4*c/(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*( e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/ 2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arc tan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1)))*e^3
Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (189) = 378\).
Time = 0.26 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.65 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
1/30*(15*b^2*sqrt((2*a^2*c*d*e^5 + sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b^4)*log(-(a^2*b^2*c^4 - a^4*d^4)*sqrt(e*x)*e^7 + (sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^7*c - (a^2*b^ 3*c^2*d - a^3*b^2*d^3)*e^5)*sqrt((2*a^2*c*d*e^5 + sqrt(-(a^3*b^2*c^4 - 2*a ^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b^4)) - 15*b^2*sqrt((2*a^2*c*d*e^5 + sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b^4)*log( -(a^2*b^2*c^4 - a^4*d^4)*sqrt(e*x)*e^7 - (sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2 *d^2 + a^5*d^4)*e^10/b^9)*b^7*c - (a^2*b^3*c^2*d - a^3*b^2*d^3)*e^5)*sqrt( (2*a^2*c*d*e^5 + sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9) *b^4)/b^4)) - 15*b^2*sqrt((2*a^2*c*d*e^5 - sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^ 2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b^4)*log(-(a^2*b^2*c^4 - a^4*d^4)*sqrt(e*x )*e^7 + (sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^7*c + (a^2*b^3*c^2*d - a^3*b^2*d^3)*e^5)*sqrt((2*a^2*c*d*e^5 - sqrt(-(a^3*b^2*c ^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b^4)) + 15*b^2*sqrt((2*a^2* c*d*e^5 - sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^4)/b ^4)*log(-(a^2*b^2*c^4 - a^4*d^4)*sqrt(e*x)*e^7 - (sqrt(-(a^3*b^2*c^4 - 2*a ^4*b*c^2*d^2 + a^5*d^4)*e^10/b^9)*b^7*c + (a^2*b^3*c^2*d - a^3*b^2*d^3)*e^ 5)*sqrt((2*a^2*c*d*e^5 - sqrt(-(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*e ^10/b^9)*b^4)/b^4)) + 4*(3*b*d*e^2*x^2 + 5*b*c*e^2*x - 15*a*d*e^2)*sqrt(e* x))/b^2
Result contains complex when optimal does not.
Time = 10.39 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.28 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x)**(5/2)*(d*x+c)/(b*x**2+a),x)
Output:
-9*a**(5/4)*d*e**(5/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*p i/4)/a**(1/4))*gamma(9/4)/(8*b**(9/4)*gamma(13/4)) + 9*I*a**(5/4)*d*e**(5/ 2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gam ma(9/4)/(8*b**(9/4)*gamma(13/4)) + 9*a**(5/4)*d*e**(5/2)*exp(-I*pi/4)*log( 1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(9/4)/(8*b**(9/4)* gamma(13/4)) - 9*I*a**(5/4)*d*e**(5/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt( x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(9/4)/(8*b**(9/4)*gamma(13/4)) + 7*a **(3/4)*c*e**(5/2)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/ 4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*gamma(11/4)) + 7*I*a**(3/4)*c*e**(5/2) *exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gam ma(7/4)/(8*b**(7/4)*gamma(11/4)) - 7*a**(3/4)*c*e**(5/2)*exp(-3*I*pi/4)*lo g(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(7/4)/(8*b**(7/4 )*gamma(11/4)) - 7*I*a**(3/4)*c*e**(5/2)*exp(-3*I*pi/4)*log(1 - b**(1/4)*s qrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*gamma(11/4)) - 9*a*d*e**(5/2)*sqrt(x)*gamma(9/4)/(2*b**2*gamma(13/4)) + 7*c*e**(5/2)*x** (3/2)*gamma(7/4)/(6*b*gamma(11/4)) + 9*d*e**(5/2)*x**(5/2)*gamma(9/4)/(10* b*gamma(13/4))
Exception generated. \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.34 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, b^{4}} + \frac {2 \, {\left (3 \, \sqrt {e x} b^{4} d e^{2} x^{2} + 5 \, \sqrt {e x} b^{4} c e^{2} x - 15 \, \sqrt {e x} a b^{3} d e^{2}\right )}}{15 \, b^{5}} \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e^2 - (a*b^3*e^2)^(3/4)*c*e)*arctan(1 /2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/b^4 + 1/2*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e^2 - (a*b^3*e^2)^(3/4)*c*e)*arctan(- 1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/b^4 + 1/4*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e^2 + (a*b^3*e^2)^(3/4)*c*e)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/b^4 - 1/4*sqrt(2)*(( a*b^3*e^2)^(1/4)*a*b*d*e^2 + (a*b^3*e^2)^(3/4)*c*e)*log(e*x - sqrt(2)*(a*e ^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/b^4 + 2/15*(3*sqrt(e*x)*b^4*d*e^2*x ^2 + 5*sqrt(e*x)*b^4*c*e^2*x - 15*sqrt(e*x)*a*b^3*d*e^2)/b^5
Time = 0.45 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.99 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=\frac {2\,d\,{\left (e\,x\right )}^{5/2}}{5\,b}+\frac {2\,c\,e\,{\left (e\,x\right )}^{3/2}}{3\,b}-\frac {2\,a\,d\,e^2\,\sqrt {e\,x}}{b^2}-\mathrm {atan}\left (\frac {a^3\,c^2\,e^8\,\sqrt {e\,x}\,\sqrt {\frac {a\,d^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^9}-\frac {c^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^8}+\frac {a^2\,c\,d\,e^5}{2\,b^4}}\,32{}\mathrm {i}}{\frac {16\,a^4\,c^3\,e^{11}}{b^2}-\frac {16\,a^4\,d^3\,e^{11}\,\sqrt {-a^3\,b^9}}{b^8}-\frac {16\,a^5\,c\,d^2\,e^{11}}{b^3}+\frac {16\,a^3\,c^2\,d\,e^{11}\,\sqrt {-a^3\,b^9}}{b^7}}-\frac {a^4\,d^2\,e^8\,\sqrt {e\,x}\,\sqrt {\frac {a\,d^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^9}-\frac {c^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^8}+\frac {a^2\,c\,d\,e^5}{2\,b^4}}\,32{}\mathrm {i}}{\frac {16\,a^4\,c^3\,e^{11}}{b}-\frac {16\,a^4\,d^3\,e^{11}\,\sqrt {-a^3\,b^9}}{b^7}-\frac {16\,a^5\,c\,d^2\,e^{11}}{b^2}+\frac {16\,a^3\,c^2\,d\,e^{11}\,\sqrt {-a^3\,b^9}}{b^6}}\right )\,\sqrt {\frac {a\,d^2\,e^5\,\sqrt {-a^3\,b^9}-b\,c^2\,e^5\,\sqrt {-a^3\,b^9}+2\,a^2\,b^5\,c\,d\,e^5}{4\,b^9}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,c^2\,e^8\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^8}-\frac {a\,d^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^9}+\frac {a^2\,c\,d\,e^5}{2\,b^4}}\,32{}\mathrm {i}}{\frac {16\,a^4\,c^3\,e^{11}}{b^2}+\frac {16\,a^4\,d^3\,e^{11}\,\sqrt {-a^3\,b^9}}{b^8}-\frac {16\,a^5\,c\,d^2\,e^{11}}{b^3}-\frac {16\,a^3\,c^2\,d\,e^{11}\,\sqrt {-a^3\,b^9}}{b^7}}-\frac {a^4\,d^2\,e^8\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^8}-\frac {a\,d^2\,e^5\,\sqrt {-a^3\,b^9}}{4\,b^9}+\frac {a^2\,c\,d\,e^5}{2\,b^4}}\,32{}\mathrm {i}}{\frac {16\,a^4\,c^3\,e^{11}}{b}+\frac {16\,a^4\,d^3\,e^{11}\,\sqrt {-a^3\,b^9}}{b^7}-\frac {16\,a^5\,c\,d^2\,e^{11}}{b^2}-\frac {16\,a^3\,c^2\,d\,e^{11}\,\sqrt {-a^3\,b^9}}{b^6}}\right )\,\sqrt {\frac {b\,c^2\,e^5\,\sqrt {-a^3\,b^9}-a\,d^2\,e^5\,\sqrt {-a^3\,b^9}+2\,a^2\,b^5\,c\,d\,e^5}{4\,b^9}}\,2{}\mathrm {i} \] Input:
int(((e*x)^(5/2)*(c + d*x))/(a + b*x^2),x)
Output:
(2*d*(e*x)^(5/2))/(5*b) - atan((a^3*c^2*e^8*(e*x)^(1/2)*((c^2*e^5*(-a^3*b^ 9)^(1/2))/(4*b^8) - (a*d^2*e^5*(-a^3*b^9)^(1/2))/(4*b^9) + (a^2*c*d*e^5)/( 2*b^4))^(1/2)*32i)/((16*a^4*c^3*e^11)/b^2 + (16*a^4*d^3*e^11*(-a^3*b^9)^(1 /2))/b^8 - (16*a^5*c*d^2*e^11)/b^3 - (16*a^3*c^2*d*e^11*(-a^3*b^9)^(1/2))/ b^7) - (a^4*d^2*e^8*(e*x)^(1/2)*((c^2*e^5*(-a^3*b^9)^(1/2))/(4*b^8) - (a*d ^2*e^5*(-a^3*b^9)^(1/2))/(4*b^9) + (a^2*c*d*e^5)/(2*b^4))^(1/2)*32i)/((16* a^4*c^3*e^11)/b + (16*a^4*d^3*e^11*(-a^3*b^9)^(1/2))/b^7 - (16*a^5*c*d^2*e ^11)/b^2 - (16*a^3*c^2*d*e^11*(-a^3*b^9)^(1/2))/b^6))*((b*c^2*e^5*(-a^3*b^ 9)^(1/2) - a*d^2*e^5*(-a^3*b^9)^(1/2) + 2*a^2*b^5*c*d*e^5)/(4*b^9))^(1/2)* 2i - atan((a^3*c^2*e^8*(e*x)^(1/2)*((a*d^2*e^5*(-a^3*b^9)^(1/2))/(4*b^9) - (c^2*e^5*(-a^3*b^9)^(1/2))/(4*b^8) + (a^2*c*d*e^5)/(2*b^4))^(1/2)*32i)/(( 16*a^4*c^3*e^11)/b^2 - (16*a^4*d^3*e^11*(-a^3*b^9)^(1/2))/b^8 - (16*a^5*c* d^2*e^11)/b^3 + (16*a^3*c^2*d*e^11*(-a^3*b^9)^(1/2))/b^7) - (a^4*d^2*e^8*( e*x)^(1/2)*((a*d^2*e^5*(-a^3*b^9)^(1/2))/(4*b^9) - (c^2*e^5*(-a^3*b^9)^(1/ 2))/(4*b^8) + (a^2*c*d*e^5)/(2*b^4))^(1/2)*32i)/((16*a^4*c^3*e^11)/b - (16 *a^4*d^3*e^11*(-a^3*b^9)^(1/2))/b^7 - (16*a^5*c*d^2*e^11)/b^2 + (16*a^3*c^ 2*d*e^11*(-a^3*b^9)^(1/2))/b^6))*((a*d^2*e^5*(-a^3*b^9)^(1/2) - b*c^2*e^5* (-a^3*b^9)^(1/2) + 2*a^2*b^5*c*d*e^5)/(4*b^9))^(1/2)*2i + (2*c*e*(e*x)^(3/ 2))/(3*b) - (2*a*d*e^2*(e*x)^(1/2))/b^2
Time = 0.23 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.18 \[ \int \frac {(e x)^{5/2} (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {e}\, e^{2} \left (30 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -30 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -30 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c +30 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -15 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c +15 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -15 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d +15 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d -120 \sqrt {x}\, a b d +40 \sqrt {x}\, b^{2} c x +24 \sqrt {x}\, b^{2} d \,x^{2}\right )}{60 b^{3}} \] Input:
int((e*x)^(5/2)*(d*x+c)/(b*x^2+a),x)
Output:
(sqrt(e)*e**2*(30*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 ) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c - 30*b**(3/4)*a**( 1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4 )*a**(1/4)*sqrt(2)))*a*d - 30*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a** (1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c + 30*b **(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt( b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*d - 15*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c + 15*b**(1/4 )*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt( b)*x)*b*c - 15*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)* sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d + 15*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt (x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d - 120*sqrt(x)*a*b *d + 40*sqrt(x)*b**2*c*x + 24*sqrt(x)*b**2*d*x**2))/(60*b**3)