Integrand size = 22, antiderivative size = 273 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-\frac {2 c}{5 a e (e x)^{5/2}}-\frac {2 d}{3 a e^2 (e x)^{3/2}}+\frac {2 b c}{a^2 e^3 \sqrt {e x}}-\frac {b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{9/4} e^{7/2}}+\frac {b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{9/4} e^{7/2}}-\frac {b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{9/4} e^{7/2}} \] Output:
-2/5*c/a/e/(e*x)^(5/2)-2/3*d/a/e^2/(e*x)^(3/2)+2*b*c/a^2/e^3/(e*x)^(1/2)-1 /2*b^(3/4)*(b^(1/2)*c-a^(1/2)*d)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1 /4)/e^(1/2))*2^(1/2)/a^(9/4)/e^(7/2)+1/2*b^(3/4)*(b^(1/2)*c-a^(1/2)*d)*arc tan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(9/4)/e^(7/2) -1/2*b^(3/4)*(b^(1/2)*c+a^(1/2)*d)*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)/a^(9/4)/e^(7/2)
Time = 0.35 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.66 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-4 \sqrt [4]{a} \left (3 a c+5 a d x-15 b c x^2\right )-15 \sqrt {2} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) x^{5/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-15 \sqrt {2} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) x^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{30 a^{9/4} (e x)^{7/2}} \] Input:
Integrate[(c + d*x)/((e*x)^(7/2)*(a + b*x^2)),x]
Output:
(x*(-4*a^(1/4)*(3*a*c + 5*a*d*x - 15*b*c*x^2) - 15*Sqrt[2]*b^(3/4)*(Sqrt[b ]*c - Sqrt[a]*d)*x^(5/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^( 1/4)*Sqrt[x])] - 15*Sqrt[2]*b^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*x^(5/2)*ArcTan h[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(30*a^(9/4)*( e*x)^(7/2))
Time = 1.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.35, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {553, 27, 553, 27, 553, 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.
| \(\displaystyle \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {2 \int -\frac {5 (a d-b c x)}{2 (e x)^{5/2} \left (b x^2+a\right )}dx}{5 a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a d-b c x}{(e x)^{5/2} \left (b x^2+a\right )}dx}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 a b (c+d x)}{2 (e x)^{3/2} \left (b x^2+a\right )}dx}{3 a e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \int \frac {c+d x}{(e x)^{3/2} \left (b x^2+a\right )}dx}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {b \left (-\frac {2 \int -\frac {a d-b c x}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\int \frac {a d-b c x}{\sqrt {e x} \left (b x^2+a\right )}dx}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \int \frac {a d e-b c e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {b \left (\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} d}{\sqrt {b}}+c\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )-\frac {1}{2} \sqrt {b} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )\right )}{a e}-\frac {2 c}{a e \sqrt {e x}}\right )}{e}-\frac {2 d}{3 e (e x)^{3/2}}}{a e}-\frac {2 c}{5 a e (e x)^{5/2}}\) |
Input:
Int[(c + d*x)/((e*x)^(7/2)*(a + b*x^2)),x]
Output:
(-2*c)/(5*a*e*(e*x)^(5/2)) + ((-2*d)/(3*e*(e*x)^(3/2)) - (b*((-2*c)/(a*e*S qrt[e*x]) + (2*(-1/2*(Sqrt[b]*(c - (Sqrt[a]*d)/Sqrt[b])*(-(ArcTan[1 - (Sqr t[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[S qrt[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))/(a* e)))/e)/(a*e)
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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
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 = 0.36 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {2 \left (-15 b c \,x^{2}+5 a d x +3 a c \right )}{15 a^{2} \sqrt {e x}\, x^{2} e^{3}}+\frac {b \left (-\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} e^{3}}\) | \(319\) |
| derivativedivides | \(-\frac {2 c}{5 a e \left (e x \right )^{\frac {5}{2}}}-\frac {2 d}{3 a \,e^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {2 b c}{a^{2} e^{3} \sqrt {e x}}-\frac {2 b \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{e^{3} a^{2}}\) | \(330\) |
| default | \(-\frac {2 c}{5 a e \left (e x \right )^{\frac {5}{2}}}-\frac {2 d}{3 a \,e^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {2 b c}{a^{2} e^{3} \sqrt {e x}}-\frac {2 b \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{e^{3} a^{2}}\) | \(330\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {5 \ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) c \sqrt {2}\, b e \,x^{2} \sqrt {e x}}{8}+\frac {5 \ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) d \sqrt {\frac {a \,e^{2}}{b}}\, \sqrt {2}\, b \,x^{2} \sqrt {e x}}{8}+\frac {5 \sqrt {2}\, b \,x^{2} \sqrt {e x}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4}+\frac {5 \sqrt {2}\, b \,x^{2} \sqrt {e x}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4}+e \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (-5 b c \,x^{2}+\frac {5}{3} a d x +a c \right )\right )}{5 \sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} e^{4} a^{2} x^{2}}\) | \(367\) |
Input:
int((d*x+c)/(e*x)^(7/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/15*(-15*b*c*x^2+5*a*d*x+3*a*c)/a^2/(e*x)^(1/2)/x^2/e^3+1/a^2*b*(-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 *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))+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 *arctan(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. 1012 vs. \(2 (194) = 388\).
Time = 0.31 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.71 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/30*(15*a^2*e^4*x^3*sqrt((a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2 *b^3*d^4)/(a^9*e^14)) + 2*b^2*c*d)/(a^4*e^7))*log(-(b^4*c^4 - a^2*b^2*d^4) *sqrt(e*x) + (a^7*c*e^11*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/( a^9*e^14)) - (a^3*b^2*c^2*d - a^4*b*d^3)*e^4)*sqrt((a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) + 2*b^2*c*d)/(a^4*e^7))) - 1 5*a^2*e^4*x^3*sqrt((a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4 )/(a^9*e^14)) + 2*b^2*c*d)/(a^4*e^7))*log(-(b^4*c^4 - a^2*b^2*d^4)*sqrt(e* x) - (a^7*c*e^11*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14 )) - (a^3*b^2*c^2*d - a^4*b*d^3)*e^4)*sqrt((a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b ^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) + 2*b^2*c*d)/(a^4*e^7))) - 15*a^2*e^ 4*x^3*sqrt(-(a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9* e^14)) - 2*b^2*c*d)/(a^4*e^7))*log(-(b^4*c^4 - a^2*b^2*d^4)*sqrt(e*x) + (a ^7*c*e^11*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) + (a ^3*b^2*c^2*d - a^4*b*d^3)*e^4)*sqrt(-(a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2 *d^2 + a^2*b^3*d^4)/(a^9*e^14)) - 2*b^2*c*d)/(a^4*e^7))) + 15*a^2*e^4*x^3* sqrt(-(a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) - 2*b^2*c*d)/(a^4*e^7))*log(-(b^4*c^4 - a^2*b^2*d^4)*sqrt(e*x) - (a^7*c*e ^11*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) + (a^3*b^2 *c^2*d - a^4*b*d^3)*e^4)*sqrt(-(a^4*e^7*sqrt(-(b^5*c^4 - 2*a*b^4*c^2*d^2 + a^2*b^3*d^4)/(a^9*e^14)) - 2*b^2*c*d)/(a^4*e^7))) - 4*(15*b*c*x^2 - 5*...
\[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(7/2)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(7/2)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(7/2)/(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 = 358, normalized size of antiderivative = 1.31 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b e^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b e^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a^{3} b e^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a^{3} b e^{5}} + \frac {2 \, {\left (15 \, b c e^{2} x^{2} - 5 \, a d e^{2} x - 3 \, a c e^{2}\right )}}{15 \, \sqrt {e x} a^{2} e^{5} x^{2}} \] Input:
integrate((d*x+c)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e - (a*b^3*e^2)^(3/4)*c)*arctan(1/2* sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^3*b*e^ 5) - 1/2*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e - (a*b^3*e^2)^(3/4)*c)*arctan( -1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^3 *b*e^5) - 1/4*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^(3/4)*c)*lo g(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^3*b*e^5) + 1 /4*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^(3/4)*c)*log(e*x - sqr t(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^3*b*e^5) + 2/15*(15*b*c *e^2*x^2 - 5*a*d*e^2*x - 3*a*c*e^2)/(sqrt(e*x)*a^2*e^5*x^2)
Time = 7.56 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.88 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-2\,\mathrm {atanh}\left (\frac {32\,a^7\,b^6\,c^2\,e^{11}\,\sqrt {e\,x}\,\sqrt {\frac {b\,c^2\,\sqrt {-a^9\,b^3}}{4\,a^9\,e^7}-\frac {d^2\,\sqrt {-a^9\,b^3}}{4\,a^8\,e^7}+\frac {b^2\,c\,d}{2\,a^4\,e^7}}}{16\,a^5\,b^7\,c^3\,e^8+16\,a^2\,b^4\,d^3\,e^8\,\sqrt {-a^9\,b^3}-16\,a^6\,b^6\,c\,d^2\,e^8-16\,a\,b^5\,c^2\,d\,e^8\,\sqrt {-a^9\,b^3}}-\frac {32\,a^8\,b^5\,d^2\,e^{11}\,\sqrt {e\,x}\,\sqrt {\frac {b\,c^2\,\sqrt {-a^9\,b^3}}{4\,a^9\,e^7}-\frac {d^2\,\sqrt {-a^9\,b^3}}{4\,a^8\,e^7}+\frac {b^2\,c\,d}{2\,a^4\,e^7}}}{16\,a^5\,b^7\,c^3\,e^8+16\,a^2\,b^4\,d^3\,e^8\,\sqrt {-a^9\,b^3}-16\,a^6\,b^6\,c\,d^2\,e^8-16\,a\,b^5\,c^2\,d\,e^8\,\sqrt {-a^9\,b^3}}\right )\,\sqrt {\frac {b\,c^2\,\sqrt {-a^9\,b^3}-a\,d^2\,\sqrt {-a^9\,b^3}+2\,a^5\,b^2\,c\,d}{4\,a^9\,e^7}}-2\,\mathrm {atanh}\left (\frac {32\,a^7\,b^6\,c^2\,e^{11}\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,\sqrt {-a^9\,b^3}}{4\,a^8\,e^7}-\frac {b\,c^2\,\sqrt {-a^9\,b^3}}{4\,a^9\,e^7}+\frac {b^2\,c\,d}{2\,a^4\,e^7}}}{16\,a^5\,b^7\,c^3\,e^8-16\,a^2\,b^4\,d^3\,e^8\,\sqrt {-a^9\,b^3}-16\,a^6\,b^6\,c\,d^2\,e^8+16\,a\,b^5\,c^2\,d\,e^8\,\sqrt {-a^9\,b^3}}-\frac {32\,a^8\,b^5\,d^2\,e^{11}\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,\sqrt {-a^9\,b^3}}{4\,a^8\,e^7}-\frac {b\,c^2\,\sqrt {-a^9\,b^3}}{4\,a^9\,e^7}+\frac {b^2\,c\,d}{2\,a^4\,e^7}}}{16\,a^5\,b^7\,c^3\,e^8-16\,a^2\,b^4\,d^3\,e^8\,\sqrt {-a^9\,b^3}-16\,a^6\,b^6\,c\,d^2\,e^8+16\,a\,b^5\,c^2\,d\,e^8\,\sqrt {-a^9\,b^3}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {-a^9\,b^3}-b\,c^2\,\sqrt {-a^9\,b^3}+2\,a^5\,b^2\,c\,d}{4\,a^9\,e^7}}-\frac {\frac {2\,c}{5\,a\,e}+\frac {2\,d\,x}{3\,a\,e}-\frac {2\,b\,c\,x^2}{a^2\,e}}{{\left (e\,x\right )}^{5/2}} \] Input:
int((c + d*x)/((e*x)^(7/2)*(a + b*x^2)),x)
Output:
- 2*atanh((32*a^7*b^6*c^2*e^11*(e*x)^(1/2)*((b*c^2*(-a^9*b^3)^(1/2))/(4*a^ 9*e^7) - (d^2*(-a^9*b^3)^(1/2))/(4*a^8*e^7) + (b^2*c*d)/(2*a^4*e^7))^(1/2) )/(16*a^5*b^7*c^3*e^8 + 16*a^2*b^4*d^3*e^8*(-a^9*b^3)^(1/2) - 16*a^6*b^6*c *d^2*e^8 - 16*a*b^5*c^2*d*e^8*(-a^9*b^3)^(1/2)) - (32*a^8*b^5*d^2*e^11*(e* x)^(1/2)*((b*c^2*(-a^9*b^3)^(1/2))/(4*a^9*e^7) - (d^2*(-a^9*b^3)^(1/2))/(4 *a^8*e^7) + (b^2*c*d)/(2*a^4*e^7))^(1/2))/(16*a^5*b^7*c^3*e^8 + 16*a^2*b^4 *d^3*e^8*(-a^9*b^3)^(1/2) - 16*a^6*b^6*c*d^2*e^8 - 16*a*b^5*c^2*d*e^8*(-a^ 9*b^3)^(1/2)))*((b*c^2*(-a^9*b^3)^(1/2) - a*d^2*(-a^9*b^3)^(1/2) + 2*a^5*b ^2*c*d)/(4*a^9*e^7))^(1/2) - 2*atanh((32*a^7*b^6*c^2*e^11*(e*x)^(1/2)*((d^ 2*(-a^9*b^3)^(1/2))/(4*a^8*e^7) - (b*c^2*(-a^9*b^3)^(1/2))/(4*a^9*e^7) + ( b^2*c*d)/(2*a^4*e^7))^(1/2))/(16*a^5*b^7*c^3*e^8 - 16*a^2*b^4*d^3*e^8*(-a^ 9*b^3)^(1/2) - 16*a^6*b^6*c*d^2*e^8 + 16*a*b^5*c^2*d*e^8*(-a^9*b^3)^(1/2)) - (32*a^8*b^5*d^2*e^11*(e*x)^(1/2)*((d^2*(-a^9*b^3)^(1/2))/(4*a^8*e^7) - (b*c^2*(-a^9*b^3)^(1/2))/(4*a^9*e^7) + (b^2*c*d)/(2*a^4*e^7))^(1/2))/(16*a ^5*b^7*c^3*e^8 - 16*a^2*b^4*d^3*e^8*(-a^9*b^3)^(1/2) - 16*a^6*b^6*c*d^2*e^ 8 + 16*a*b^5*c^2*d*e^8*(-a^9*b^3)^(1/2)))*((a*d^2*(-a^9*b^3)^(1/2) - b*c^2 *(-a^9*b^3)^(1/2) + 2*a^5*b^2*c*d)/(4*a^9*e^7))^(1/2) - ((2*c)/(5*a*e) + ( 2*d*x)/(3*a*e) - (2*b*c*x^2)/(a^2*e))/(e*x)^(5/2)
Time = 0.27 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.31 \[ \int \frac {c+d x}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (-30 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,x^{2}+30 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d \,x^{2}+30 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,x^{2}-30 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d \,x^{2}+15 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c \,x^{2}-15 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c \,x^{2}+15 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d \,x^{2}-15 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d \,x^{2}-24 a^{2} c -40 a^{2} d x +120 a b c \,x^{2}\right )}{60 \sqrt {x}\, a^{3} e^{4} x^{2}} \] Input:
int((d*x+c)/(e*x)^(7/2)/(b*x^2+a),x)
Output:
(sqrt(e)*( - 30*sqrt(x)*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*x**2 + 30*sq rt(x)*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*x**2 + 30*sqrt(x)*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*x**2 - 30*sqrt(x)*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*x**2 + 15*sqrt(x)*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*x**2 - 15*sqrt(x)*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*x**2 + 15*sqrt(x)*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*x**2 - 15*sqrt(x)*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*x**2 - 24*a**2*c - 40*a**2*d*x + 120*a*b*c*x**2))/(60*sqrt(x)*a**3*e **4*x**2)