3.377 \(\int \frac{1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=359 \[ -\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}-\frac{e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{c (2 c d-b e)}{b^2 d (b+c x) (d+e x)^{5/2} (c d-b e)}-\frac{e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{1}{b d x (b+c x) (d+e x)^{5/2}} \]

[Out]

-(e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5
/2)) - (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*(d + e*x)^(5/2)) - 1/(b*d*
x*(b + c*x)*(d + e*x)^(5/2)) - (e*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 7*b^2*e
^2))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*c^4*d^4 - 4*b*c^3*d^3*e +
26*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d
+ e*x]) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) - (c^(
9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*
d - b*e)^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.91092, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}-\frac{e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{c (2 c d-b e)}{b^2 d (b+c x) (d+e x)^{5/2} (c d-b e)}-\frac{e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{1}{b d x (b+c x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5
/2)) - (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*(d + e*x)^(5/2)) - 1/(b*d*
x*(b + c*x)*(d + e*x)^(5/2)) - (e*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 7*b^2*e
^2))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*c^4*d^4 - 4*b*c^3*d^3*e +
26*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d
+ e*x]) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) - (c^(
9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*
d - b*e)^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 2.173, size = 251, normalized size = 0.7 \[ -\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}+\sqrt{d+e x} \left (-\frac{c^5}{b^2 (b+c x) (c d-b e)^4}-\frac{2 e^3 \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{d^4 (d+e x) (c d-b e)^4}-\frac{1}{b^2 d^4 x}+\frac{4 e^3 (b e-2 c d)}{3 d^3 (d+e x)^2 (c d-b e)^3}-\frac{2 e^3}{5 d^2 (d+e x)^3 (c d-b e)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

Sqrt[d + e*x]*(-(1/(b^2*d^4*x)) - c^5/(b^2*(c*d - b*e)^4*(b + c*x)) - (2*e^3)/(5
*d^2*(c*d - b*e)^2*(d + e*x)^3) + (4*e^3*(-2*c*d + b*e))/(3*d^3*(c*d - b*e)^3*(d
+ e*x)^2) - (2*e^3*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))/(d^4*(c*d - b*e)^4*(d
+ e*x))) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) - (c^
(9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c
*d - b*e)^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.042, size = 364, normalized size = 1. \[ -{\frac{2\,{e}^{3}}{5\,{d}^{2} \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,{e}^{4}b}{3\,{d}^{3} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{3}c}{3\,{d}^{2} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-6\,{\frac{{e}^{5}{b}^{2}}{{d}^{4} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+20\,{\frac{{e}^{4}bc}{{d}^{3} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-20\,{\frac{{e}^{3}{c}^{2}}{{d}^{2} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-{\frac{e{c}^{5}}{{b}^{2} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) }\sqrt{ex+d}}-11\,{\frac{e{c}^{5}}{{b}^{2} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{6}d}{{b}^{3} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{d}^{4}{b}^{2}x}\sqrt{ex+d}}+7\,{\frac{e}{{d}^{9/2}{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{c}{{d}^{7/2}{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x)

[Out]

-2/5*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)-4/3*e^4/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)*b+8
/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*c-6*e^5/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*b^2+2
0*e^4/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*b*c-20*e^3/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*c^2
-e*c^5/b^2/(b*e-c*d)^4*(e*x+d)^(1/2)/(c*e*x+b*e)-11*e*c^5/b^2/(b*e-c*d)^4/((b*e-
c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+4*c^6/b^3/(b*e-c*d)^4/
((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d-1/d^4/b^2*(e*x
+d)^(1/2)/x+7*e/d^(9/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))+4/d^(7/2)/b^3*arctanh
((e*x+d)^(1/2)/d^(1/2))*c

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 7.26719, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/30*(15*((4*c^6*d^5*e^2 - 11*b*c^5*d^4*e^3)*x^4 + (8*c^6*d^6*e - 18*b*c^5*d^5
*e^2 - 11*b^2*c^4*d^4*e^3)*x^3 + (4*c^6*d^7 - 3*b*c^5*d^6*e - 22*b^2*c^4*d^5*e^2
)*x^2 + (4*b*c^5*d^7 - 11*b^2*c^4*d^6*e)*x)*sqrt(e*x + d)*sqrt(d)*sqrt(c/(c*d -
b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e))
)/(c*x + b)) - 15*((4*c^6*d^5*e^2 - 9*b*c^5*d^4*e^3 - 4*b^2*c^4*d^3*e^4 + 26*b^3
*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 + (8*c^6*d^6*e - 14*b*c^5*d^5
*e^2 - 17*b^2*c^4*d^4*e^3 + 48*b^3*c^3*d^3*e^4 - 22*b^4*c^2*d^2*e^5 - 10*b^5*c*d
*e^6 + 7*b^6*e^7)*x^3 + (4*c^6*d^7 - b*c^5*d^6*e - 22*b^2*c^4*d^5*e^2 + 18*b^3*c
^3*d^4*e^3 + 28*b^4*c^2*d^3*e^4 - 41*b^5*c*d^2*e^5 + 14*b^6*d*e^6)*x^2 + (4*b*c^
5*d^7 - 9*b^2*c^4*d^6*e - 4*b^3*c^3*d^5*e^2 + 26*b^4*c^2*d^4*e^3 - 24*b^5*c*d^3*
e^4 + 7*b^6*d^2*e^5)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)
*d)/x) + 2*(15*b^2*c^4*d^7 - 60*b^3*c^3*d^6*e + 90*b^4*c^2*d^5*e^2 - 60*b^5*c*d^
4*e^3 + 15*b^6*d^3*e^4 + 15*(2*b*c^5*d^4*e^3 - 4*b^2*c^4*d^3*e^4 + 26*b^3*c^3*d^
2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 + 5*(18*b*c^5*d^5*e^2 - 33*b^2*c^4*d
^4*e^3 + 170*b^3*c^3*d^3*e^4 - 90*b^4*c^2*d^2*e^5 - 23*b^5*c*d*e^6 + 21*b^6*e^7)
*x^3 + (90*b*c^5*d^6*e - 135*b^2*c^4*d^5*e^2 + 436*b^3*c^3*d^4*e^3 + 358*b^4*c^2
*d^3*e^4 - 679*b^5*c*d^2*e^5 + 245*b^6*d*e^6)*x^2 + (30*b*c^5*d^7 - 15*b^2*c^4*d
^6*e - 90*b^3*c^3*d^5*e^2 + 556*b^4*c^2*d^4*e^3 - 537*b^5*c*d^3*e^4 + 161*b^6*d^
2*e^5)*x)*sqrt(d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 -
4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 +
8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3 + (b^
3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*d
^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e^
2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sqrt(d)), -1/30*(30*((4*c^6*
d^5*e^2 - 11*b*c^5*d^4*e^3)*x^4 + (8*c^6*d^6*e - 18*b*c^5*d^5*e^2 - 11*b^2*c^4*d
^4*e^3)*x^3 + (4*c^6*d^7 - 3*b*c^5*d^6*e - 22*b^2*c^4*d^5*e^2)*x^2 + (4*b*c^5*d^
7 - 11*b^2*c^4*d^6*e)*x)*sqrt(e*x + d)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d
- b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - 15*((4*c^6*d^5*e^2 - 9*b*c^5*d
^4*e^3 - 4*b^2*c^4*d^3*e^4 + 26*b^3*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7
)*x^4 + (8*c^6*d^6*e - 14*b*c^5*d^5*e^2 - 17*b^2*c^4*d^4*e^3 + 48*b^3*c^3*d^3*e^
4 - 22*b^4*c^2*d^2*e^5 - 10*b^5*c*d*e^6 + 7*b^6*e^7)*x^3 + (4*c^6*d^7 - b*c^5*d^
6*e - 22*b^2*c^4*d^5*e^2 + 18*b^3*c^3*d^4*e^3 + 28*b^4*c^2*d^3*e^4 - 41*b^5*c*d^
2*e^5 + 14*b^6*d*e^6)*x^2 + (4*b*c^5*d^7 - 9*b^2*c^4*d^6*e - 4*b^3*c^3*d^5*e^2 +
26*b^4*c^2*d^4*e^3 - 24*b^5*c*d^3*e^4 + 7*b^6*d^2*e^5)*x)*sqrt(e*x + d)*log(((e
*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x) + 2*(15*b^2*c^4*d^7 - 60*b^3*c^3*d^6*e
+ 90*b^4*c^2*d^5*e^2 - 60*b^5*c*d^4*e^3 + 15*b^6*d^3*e^4 + 15*(2*b*c^5*d^4*e^3
- 4*b^2*c^4*d^3*e^4 + 26*b^3*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 +
5*(18*b*c^5*d^5*e^2 - 33*b^2*c^4*d^4*e^3 + 170*b^3*c^3*d^3*e^4 - 90*b^4*c^2*d^2
*e^5 - 23*b^5*c*d*e^6 + 21*b^6*e^7)*x^3 + (90*b*c^5*d^6*e - 135*b^2*c^4*d^5*e^2
+ 436*b^3*c^3*d^4*e^3 + 358*b^4*c^2*d^3*e^4 - 679*b^5*c*d^2*e^5 + 245*b^6*d*e^6)
*x^2 + (30*b*c^5*d^7 - 15*b^2*c^4*d^6*e - 90*b^3*c^3*d^5*e^2 + 556*b^4*c^2*d^4*e
^3 - 537*b^5*c*d^3*e^4 + 161*b^6*d^2*e^5)*x)*sqrt(d))/(((b^3*c^5*d^8*e^2 - 4*b^4
*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b
^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7
*c*d^5*e^5 + b^8*d^4*e^6)*x^3 + (b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*
e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 -
4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*
x + d)*sqrt(d)), -1/30*(15*((4*c^6*d^5*e^2 - 11*b*c^5*d^4*e^3)*x^4 + (8*c^6*d^6*
e - 18*b*c^5*d^5*e^2 - 11*b^2*c^4*d^4*e^3)*x^3 + (4*c^6*d^7 - 3*b*c^5*d^6*e - 22
*b^2*c^4*d^5*e^2)*x^2 + (4*b*c^5*d^7 - 11*b^2*c^4*d^6*e)*x)*sqrt(e*x + d)*sqrt(-
d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sq
rt(c/(c*d - b*e)))/(c*x + b)) + 30*((4*c^6*d^5*e^2 - 9*b*c^5*d^4*e^3 - 4*b^2*c^4
*d^3*e^4 + 26*b^3*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 + (8*c^6*d^6
*e - 14*b*c^5*d^5*e^2 - 17*b^2*c^4*d^4*e^3 + 48*b^3*c^3*d^3*e^4 - 22*b^4*c^2*d^2
*e^5 - 10*b^5*c*d*e^6 + 7*b^6*e^7)*x^3 + (4*c^6*d^7 - b*c^5*d^6*e - 22*b^2*c^4*d
^5*e^2 + 18*b^3*c^3*d^4*e^3 + 28*b^4*c^2*d^3*e^4 - 41*b^5*c*d^2*e^5 + 14*b^6*d*e
^6)*x^2 + (4*b*c^5*d^7 - 9*b^2*c^4*d^6*e - 4*b^3*c^3*d^5*e^2 + 26*b^4*c^2*d^4*e^
3 - 24*b^5*c*d^3*e^4 + 7*b^6*d^2*e^5)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*s
qrt(-d))) + 2*(15*b^2*c^4*d^7 - 60*b^3*c^3*d^6*e + 90*b^4*c^2*d^5*e^2 - 60*b^5*c
*d^4*e^3 + 15*b^6*d^3*e^4 + 15*(2*b*c^5*d^4*e^3 - 4*b^2*c^4*d^3*e^4 + 26*b^3*c^3
*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 + 5*(18*b*c^5*d^5*e^2 - 33*b^2*c^
4*d^4*e^3 + 170*b^3*c^3*d^3*e^4 - 90*b^4*c^2*d^2*e^5 - 23*b^5*c*d*e^6 + 21*b^6*e
^7)*x^3 + (90*b*c^5*d^6*e - 135*b^2*c^4*d^5*e^2 + 436*b^3*c^3*d^4*e^3 + 358*b^4*
c^2*d^3*e^4 - 679*b^5*c*d^2*e^5 + 245*b^6*d*e^6)*x^2 + (30*b*c^5*d^7 - 15*b^2*c^
4*d^6*e - 90*b^3*c^3*d^5*e^2 + 556*b^4*c^2*d^4*e^3 - 537*b^5*c*d^3*e^4 + 161*b^6
*d^2*e^5)*x)*sqrt(-d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^
4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^
2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3 +
(b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7
*c*d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^
8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sqrt(-d)), -1/15*(15*((4
*c^6*d^5*e^2 - 11*b*c^5*d^4*e^3)*x^4 + (8*c^6*d^6*e - 18*b*c^5*d^5*e^2 - 11*b^2*
c^4*d^4*e^3)*x^3 + (4*c^6*d^7 - 3*b*c^5*d^6*e - 22*b^2*c^4*d^5*e^2)*x^2 + (4*b*c
^5*d^7 - 11*b^2*c^4*d^6*e)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan
(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 15*((4*c^6*d^5*e^2 - 9*b
*c^5*d^4*e^3 - 4*b^2*c^4*d^3*e^4 + 26*b^3*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5
*c*e^7)*x^4 + (8*c^6*d^6*e - 14*b*c^5*d^5*e^2 - 17*b^2*c^4*d^4*e^3 + 48*b^3*c^3*
d^3*e^4 - 22*b^4*c^2*d^2*e^5 - 10*b^5*c*d*e^6 + 7*b^6*e^7)*x^3 + (4*c^6*d^7 - b*
c^5*d^6*e - 22*b^2*c^4*d^5*e^2 + 18*b^3*c^3*d^4*e^3 + 28*b^4*c^2*d^3*e^4 - 41*b^
5*c*d^2*e^5 + 14*b^6*d*e^6)*x^2 + (4*b*c^5*d^7 - 9*b^2*c^4*d^6*e - 4*b^3*c^3*d^5
*e^2 + 26*b^4*c^2*d^4*e^3 - 24*b^5*c*d^3*e^4 + 7*b^6*d^2*e^5)*x)*sqrt(e*x + d)*a
rctan(d/(sqrt(e*x + d)*sqrt(-d))) + (15*b^2*c^4*d^7 - 60*b^3*c^3*d^6*e + 90*b^4*
c^2*d^5*e^2 - 60*b^5*c*d^4*e^3 + 15*b^6*d^3*e^4 + 15*(2*b*c^5*d^4*e^3 - 4*b^2*c^
4*d^3*e^4 + 26*b^3*c^3*d^2*e^5 - 24*b^4*c^2*d*e^6 + 7*b^5*c*e^7)*x^4 + 5*(18*b*c
^5*d^5*e^2 - 33*b^2*c^4*d^4*e^3 + 170*b^3*c^3*d^3*e^4 - 90*b^4*c^2*d^2*e^5 - 23*
b^5*c*d*e^6 + 21*b^6*e^7)*x^3 + (90*b*c^5*d^6*e - 135*b^2*c^4*d^5*e^2 + 436*b^3*
c^3*d^4*e^3 + 358*b^4*c^2*d^3*e^4 - 679*b^5*c*d^2*e^5 + 245*b^6*d*e^6)*x^2 + (30
*b*c^5*d^7 - 15*b^2*c^4*d^6*e - 90*b^3*c^3*d^5*e^2 + 556*b^4*c^2*d^4*e^3 - 537*b
^5*c*d^3*e^4 + 161*b^6*d^2*e^5)*x)*sqrt(-d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*
e^3 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^
9*e - 7*b^4*c^4*d^8*e^2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^
5 + b^8*d^4*e^6)*x^3 + (b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b
^6*c^2*d^7*e^3 - 7*b^7*c*d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^
3*d^9*e + 6*b^6*c^2*d^8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sq
rt(-d))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239151, size = 868, normalized size = 2.42 \[ \frac{{\left (4 \, c^{6} d - 11 \, b c^{5} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{4} e - 2 \, \sqrt{x e + d} c^{5} d^{5} e - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{3} e^{2} + 5 \, \sqrt{x e + d} b c^{4} d^{4} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} e^{3} - 10 \, \sqrt{x e + d} b^{2} c^{3} d^{3} e^{3} - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d e^{4} + 10 \, \sqrt{x e + d} b^{3} c^{2} d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c e^{5} - 5 \, \sqrt{x e + d} b^{4} c d e^{5} + \sqrt{x e + d} b^{5} e^{6}}{{\left (b^{2} c^{4} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + b^{6} d^{4} e^{4}\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}} - \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} c^{2} d^{2} e^{3} + 20 \,{\left (x e + d\right )} c^{2} d^{3} e^{3} + 3 \, c^{2} d^{4} e^{3} - 150 \,{\left (x e + d\right )}^{2} b c d e^{4} - 30 \,{\left (x e + d\right )} b c d^{2} e^{4} - 6 \, b c d^{3} e^{4} + 45 \,{\left (x e + d\right )}^{2} b^{2} e^{5} + 10 \,{\left (x e + d\right )} b^{2} d e^{5} + 3 \, b^{2} d^{2} e^{5}\right )}}{15 \,{\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} - \frac{{\left (4 \, c d + 7 \, b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

(4*c^6*d - 11*b*c^5*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^4*d^
4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*sqrt(-c^2*d +
b*c*e)) - (2*(x*e + d)^(3/2)*c^5*d^4*e - 2*sqrt(x*e + d)*c^5*d^5*e - 4*(x*e + d
)^(3/2)*b*c^4*d^3*e^2 + 5*sqrt(x*e + d)*b*c^4*d^4*e^2 + 6*(x*e + d)^(3/2)*b^2*c^
3*d^2*e^3 - 10*sqrt(x*e + d)*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*b^3*c^2*d*e^4 +
10*sqrt(x*e + d)*b^3*c^2*d^2*e^4 + (x*e + d)^(3/2)*b^4*c*e^5 - 5*sqrt(x*e + d)*
b^4*c*d*e^5 + sqrt(x*e + d)*b^5*e^6)/((b^2*c^4*d^8 - 4*b^3*c^3*d^7*e + 6*b^4*c^2
*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d
^2 + (x*e + d)*b*e - b*d*e)) - 2/15*(150*(x*e + d)^2*c^2*d^2*e^3 + 20*(x*e + d)*
c^2*d^3*e^3 + 3*c^2*d^4*e^3 - 150*(x*e + d)^2*b*c*d*e^4 - 30*(x*e + d)*b*c*d^2*e
^4 - 6*b*c*d^3*e^4 + 45*(x*e + d)^2*b^2*e^5 + 10*(x*e + d)*b^2*d*e^5 + 3*b^2*d^2
*e^5)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*
e^4)*(x*e + d)^(5/2)) - (4*c*d + 7*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt
(-d)*d^4)