Integrand size = 22, antiderivative size = 248 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=\frac {2 c e \sqrt {e x}}{b}+\frac {2 d (e x)^{3/2}}{3 b}+\frac {\sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} b^{7/4}} \] Output:
2*c*e*(e*x)^(1/2)/b+2/3*d*(e*x)^(3/2)/b+1/2*a^(1/4)*(b^(1/2)*c+a^(1/2)*d)* e^(3/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(7 /4)-1/2*a^(1/4)*(b^(1/2)*c+a^(1/2)*d)*e^(3/2)*arctan(1+2^(1/2)*b^(1/4)*(e* x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(7/4)-1/2*a^(1/4)*(b^(1/2)*c-a^(1/2)*d )*e^(3/2)*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^(7/4)
Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=\frac {(e x)^{3/2} \left (4 b^{3/4} \sqrt {x} (3 c+d x)+3 \sqrt {2} \sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \sqrt [4]{a} \left (-\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{6 b^{7/4} x^{3/2}} \] Input:
Integrate[((e*x)^(3/2)*(c + d*x))/(a + b*x^2),x]
Output:
((e*x)^(3/2)*(4*b^(3/4)*Sqrt[x]*(3*c + d*x) + 3*Sqrt[2]*a^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x ])] + 3*Sqrt[2]*a^(1/4)*(-(Sqrt[b]*c) + Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a^(1/4 )*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(6*b^(7/4)*x^(3/2))
Time = 0.97 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.34, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {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.
| \(\displaystyle \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {2 e \int \frac {3 \sqrt {e x} (a d-b c x)}{2 \left (b x^2+a\right )}dx}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \int \frac {\sqrt {e x} (a d-b c x)}{b x^2+a}dx}{b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (-\frac {2 e \int -\frac {a b (c+d x)}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{b}-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (a e \int \frac {c+d x}{\sqrt {e x} \left (b x^2+a\right )}dx-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \int \frac {c e+d x e}{b x^2 e^2+a e^2}d\sqrt {e x}-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-2 c \sqrt {e x}\right )}{b}\) |
Input:
Int[((e*x)^(3/2)*(c + d*x))/(a + b*x^2),x]
Output:
(2*d*(e*x)^(3/2))/(3*b) - (e*(-2*c*Sqrt[e*x] + 2*a*e*((((Sqrt[b]*c)/Sqrt[a ] + d)*(-(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])))/(2*Sqrt[b]) + (((Sqrt[b ]*c)/Sqrt[a] - d)*(-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*Sqrt[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 = 0.37 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(-\frac {\left (b \sqrt {2}\, c \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\frac {a \,e^{2}}{b}}-8 \sqrt {e x}\, b \left (\frac {d x}{3}+c \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+d e \sqrt {2}\, a \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )\right ) e}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{2}}\) | \(305\) |
| risch | \(\frac {2 \left (d x +3 c \right ) x \,e^{2}}{3 b \sqrt {e x}}-\frac {a \left (\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e a}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right ) e^{2}}{b}\) | \(314\) |
| derivativedivides | \(\frac {\frac {2 d \left (e x \right )^{\frac {3}{2}}}{3}+2 c e \sqrt {e x}}{b}-\frac {2 a \,e^{2} \left (\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b}\) | \(315\) |
| default | \(\frac {\frac {2 d \left (e x \right )^{\frac {3}{2}}}{3}+2 c e \sqrt {e x}}{b}-\frac {2 a \,e^{2} \left (\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b}\) | \(315\) |
Input:
int((e*x)^(3/2)*(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/4*(b*2^(1/2)*c*(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*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e *x)^(1/2)-1))*(a*e^2/b)^(1/2)-8*(e*x)^(1/2)*b*(1/3*d*x+c)*(a*e^2/b)^(1/4)+ d*e*2^(1/2)*a*(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*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^ (1/2)-1)))*e/(a*e^2/b)^(1/4)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 886 vs. \(2 (173) = 346\).
Time = 0.15 (sec) , antiderivative size = 886, normalized size of antiderivative = 3.57 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x)^(3/2)*(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
1/6*(3*b*sqrt(-(2*a*c*d*e^3 + b^3*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3 *d^4)*e^6/b^7))/b^3)*log(-(b^2*c^4 - a^2*d^4)*sqrt(e*x)*e^4 + (b^5*d*sqrt( -(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7) + (b^3*c^3 - a*b^2*c*d^2 )*e^3)*sqrt(-(2*a*c*d*e^3 + b^3*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d ^4)*e^6/b^7))/b^3)) - 3*b*sqrt(-(2*a*c*d*e^3 + b^3*sqrt(-(a*b^2*c^4 - 2*a^ 2*b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)*log(-(b^2*c^4 - a^2*d^4)*sqrt(e*x)*e ^4 - (b^5*d*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7) + (b^3* c^3 - a*b^2*c*d^2)*e^3)*sqrt(-(2*a*c*d*e^3 + b^3*sqrt(-(a*b^2*c^4 - 2*a^2* b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)) - 3*b*sqrt(-(2*a*c*d*e^3 - b^3*sqrt(- (a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)*log(-(b^2*c^4 - a^2 *d^4)*sqrt(e*x)*e^4 + (b^5*d*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4) *e^6/b^7) - (b^3*c^3 - a*b^2*c*d^2)*e^3)*sqrt(-(2*a*c*d*e^3 - b^3*sqrt(-(a *b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)) + 3*b*sqrt(-(2*a*c*d *e^3 - b^3*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)*lo g(-(b^2*c^4 - a^2*d^4)*sqrt(e*x)*e^4 - (b^5*d*sqrt(-(a*b^2*c^4 - 2*a^2*b*c ^2*d^2 + a^3*d^4)*e^6/b^7) - (b^3*c^3 - a*b^2*c*d^2)*e^3)*sqrt(-(2*a*c*d*e ^3 - b^3*sqrt(-(a*b^2*c^4 - 2*a^2*b*c^2*d^2 + a^3*d^4)*e^6/b^7))/b^3)) + 4 *(d*e*x + 3*c*e)*sqrt(e*x))/b
Result contains complex when optimal does not.
Time = 9.51 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.34 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x)**(3/2)*(d*x+c)/(b*x**2+a),x)
Output:
7*a**(3/4)*d*e**(3/2)*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)*d*e**(3 /2)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))* gamma(7/4)/(8*b**(7/4)*gamma(11/4)) - 7*a**(3/4)*d*e**(3/2)*exp(-3*I*pi/4) *log(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)*d*e**(3/2)*exp(-3*I*pi/4)*log(1 - b**(1/4 )*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*gamma(11/4) ) + 5*a**(1/4)*c*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar( I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) - 5*I*a**(1/4)*c*e**( 3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*g amma(5/4)/(8*b**(5/4)*gamma(9/4)) - 5*a**(1/4)*c*e**(3/2)*exp(-I*pi/4)*log (1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4) *gamma(9/4)) + 5*I*a**(1/4)*c*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt( x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) + 5*c* e**(3/2)*sqrt(x)*gamma(5/4)/(2*b*gamma(9/4)) + 7*d*e**(3/2)*x**(3/2)*gamma (7/4)/(6*b*gamma(11/4))
Exception generated. \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(3/2)*(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
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (173) = 346\).
Time = 0.14 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.40 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=-\frac {1}{12} \, e {\left (\frac {6 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} e} + \frac {6 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} e} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{b^{4} e} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{b^{4} e} - \frac {8 \, {\left (\sqrt {e x} b^{2} d e^{3} x + 3 \, \sqrt {e x} b^{2} c e^{3}\right )}}{b^{3} e^{3}}\right )} \] Input:
integrate((e*x)^(3/2)*(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
-1/12*e*(6*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e + (a*b^3*e^2)^(3/4)*d)*arcta n(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(b^ 4*e) + 6*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e + (a*b^3*e^2)^(3/4)*d)*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 *e) + 3*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e - (a*b^3*e^2)^(3/4)*d)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(b^4*e) - 3*sqrt(2)*( (a*b^3*e^2)^(1/4)*b^2*c*e - (a*b^3*e^2)^(3/4)*d)*log(e*x - sqrt(2)*(a*e^2/ b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(b^4*e) - 8*(sqrt(e*x)*b^2*d*e^3*x + 3 *sqrt(e*x)*b^2*c*e^3)/(b^3*e^3))
Time = 7.32 (sec) , antiderivative size = 728, normalized size of antiderivative = 2.94 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=\frac {2\,d\,{\left (e\,x\right )}^{3/2}}{3\,b}+\frac {2\,c\,e\,\sqrt {e\,x}}{b}-\mathrm {atan}\left (\frac {a^3\,d^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {a\,d^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^7}-\frac {a\,c\,d\,e^3}{2\,b^3}-\frac {c^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^6}}\,32{}\mathrm {i}}{\frac {16\,a^4\,d^3\,e^8}{b^2}-\frac {16\,a^2\,c^3\,e^8\,\sqrt {-a\,b^7}}{b^4}-\frac {16\,a^3\,c^2\,d\,e^8}{b}+\frac {16\,a^3\,c\,d^2\,e^8\,\sqrt {-a\,b^7}}{b^5}}-\frac {a^2\,b\,c^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {a\,d^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^7}-\frac {a\,c\,d\,e^3}{2\,b^3}-\frac {c^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^6}}\,32{}\mathrm {i}}{\frac {16\,a^4\,d^3\,e^8}{b^2}-\frac {16\,a^2\,c^3\,e^8\,\sqrt {-a\,b^7}}{b^4}-\frac {16\,a^3\,c^2\,d\,e^8}{b}+\frac {16\,a^3\,c\,d^2\,e^8\,\sqrt {-a\,b^7}}{b^5}}\right )\,\sqrt {-\frac {b\,c^2\,e^3\,\sqrt {-a\,b^7}-a\,d^2\,e^3\,\sqrt {-a\,b^7}+2\,a\,b^4\,c\,d\,e^3}{4\,b^7}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,d^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^6}-\frac {a\,c\,d\,e^3}{2\,b^3}-\frac {a\,d^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^7}}\,32{}\mathrm {i}}{\frac {16\,a^4\,d^3\,e^8}{b^2}+\frac {16\,a^2\,c^3\,e^8\,\sqrt {-a\,b^7}}{b^4}-\frac {16\,a^3\,c^2\,d\,e^8}{b}-\frac {16\,a^3\,c\,d^2\,e^8\,\sqrt {-a\,b^7}}{b^5}}-\frac {a^2\,b\,c^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^6}-\frac {a\,c\,d\,e^3}{2\,b^3}-\frac {a\,d^2\,e^3\,\sqrt {-a\,b^7}}{4\,b^7}}\,32{}\mathrm {i}}{\frac {16\,a^4\,d^3\,e^8}{b^2}+\frac {16\,a^2\,c^3\,e^8\,\sqrt {-a\,b^7}}{b^4}-\frac {16\,a^3\,c^2\,d\,e^8}{b}-\frac {16\,a^3\,c\,d^2\,e^8\,\sqrt {-a\,b^7}}{b^5}}\right )\,\sqrt {-\frac {a\,d^2\,e^3\,\sqrt {-a\,b^7}-b\,c^2\,e^3\,\sqrt {-a\,b^7}+2\,a\,b^4\,c\,d\,e^3}{4\,b^7}}\,2{}\mathrm {i} \] Input:
int(((e*x)^(3/2)*(c + d*x))/(a + b*x^2),x)
Output:
(2*d*(e*x)^(3/2))/(3*b) - atan((a^3*d^2*e^6*(e*x)^(1/2)*((c^2*e^3*(-a*b^7) ^(1/2))/(4*b^6) - (a*c*d*e^3)/(2*b^3) - (a*d^2*e^3*(-a*b^7)^(1/2))/(4*b^7) )^(1/2)*32i)/((16*a^4*d^3*e^8)/b^2 + (16*a^2*c^3*e^8*(-a*b^7)^(1/2))/b^4 - (16*a^3*c^2*d*e^8)/b - (16*a^3*c*d^2*e^8*(-a*b^7)^(1/2))/b^5) - (a^2*b*c^ 2*e^6*(e*x)^(1/2)*((c^2*e^3*(-a*b^7)^(1/2))/(4*b^6) - (a*c*d*e^3)/(2*b^3) - (a*d^2*e^3*(-a*b^7)^(1/2))/(4*b^7))^(1/2)*32i)/((16*a^4*d^3*e^8)/b^2 + ( 16*a^2*c^3*e^8*(-a*b^7)^(1/2))/b^4 - (16*a^3*c^2*d*e^8)/b - (16*a^3*c*d^2* e^8*(-a*b^7)^(1/2))/b^5))*(-(a*d^2*e^3*(-a*b^7)^(1/2) - b*c^2*e^3*(-a*b^7) ^(1/2) + 2*a*b^4*c*d*e^3)/(4*b^7))^(1/2)*2i - atan((a^3*d^2*e^6*(e*x)^(1/2 )*((a*d^2*e^3*(-a*b^7)^(1/2))/(4*b^7) - (a*c*d*e^3)/(2*b^3) - (c^2*e^3*(-a *b^7)^(1/2))/(4*b^6))^(1/2)*32i)/((16*a^4*d^3*e^8)/b^2 - (16*a^2*c^3*e^8*( -a*b^7)^(1/2))/b^4 - (16*a^3*c^2*d*e^8)/b + (16*a^3*c*d^2*e^8*(-a*b^7)^(1/ 2))/b^5) - (a^2*b*c^2*e^6*(e*x)^(1/2)*((a*d^2*e^3*(-a*b^7)^(1/2))/(4*b^7) - (a*c*d*e^3)/(2*b^3) - (c^2*e^3*(-a*b^7)^(1/2))/(4*b^6))^(1/2)*32i)/((16* a^4*d^3*e^8)/b^2 - (16*a^2*c^3*e^8*(-a*b^7)^(1/2))/b^4 - (16*a^3*c^2*d*e^8 )/b + (16*a^3*c*d^2*e^8*(-a*b^7)^(1/2))/b^5))*(-(b*c^2*e^3*(-a*b^7)^(1/2) - a*d^2*e^3*(-a*b^7)^(1/2) + 2*a*b^4*c*d*e^3)/(4*b^7))^(1/2)*2i + (2*c*e*( e*x)^(1/2))/b
Time = 0.21 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.21 \[ \int \frac {(e x)^{3/2} (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {e}\, e \left (6 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +6 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -6 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -6 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -3 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d +3 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d +3 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -3 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c +24 \sqrt {x}\, b c +8 \sqrt {x}\, b d x \right )}{12 b^{2}} \] Input:
int((e*x)^(3/2)*(d*x+c)/(b*x^2+a),x)
Output:
(sqrt(e)*e*(6*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)))*d + 6*b**(3/4)*a**(1/4)*sq rt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1 /4)*sqrt(2)))*c - 6*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)))*d - 6*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)))*c - 3*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4 )*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d + 3*b**(1/4)*a**(3/4)*sqrt(2)* log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d + 3*b**(3/4 )*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq rt(b)*x)*c - 3*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqr t(2) + sqrt(a) + sqrt(b)*x)*c + 24*sqrt(x)*b*c + 8*sqrt(x)*b*d*x))/(12*b** 2)