3.378 \(\int \frac{(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=312 \[ \frac{d (d+e x)^{5/2} (8 c d-11 b e)}{4 b^2 x (b+c x)^2}-\frac{3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac{3 \sqrt{d+e x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2 (b+c x)}+\frac{(d+e x)^{3/2} (c d-b e) \left (2 b^2 e^2-17 b c d e+12 c^2 d^2\right )}{4 b^3 c (b+c x)^2}-\frac{d (d+e x)^{7/2}}{2 b x^2 (b+c x)^2} \]

[Out]

(3*(c*d - b*e)*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4
*b^4*c^2*(b + c*x)) + ((c*d - b*e)*(12*c^2*d^2 - 17*b*c*d*e + 2*b^2*e^2)*(d + e*
x)^(3/2))/(4*b^3*c*(b + c*x)^2) + (d*(8*c*d - 11*b*e)*(d + e*x)^(5/2))/(4*b^2*x*
(b + c*x)^2) - (d*(d + e*x)^(7/2))/(2*b*x^2*(b + c*x)^2) - (3*d^(5/2)*(16*c^2*d^
2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d -
b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/S
qrt[c*d - b*e]])/(4*b^5*c^(5/2))

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Rubi [A]  time = 1.216, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{d (d+e x)^{5/2} (8 c d-11 b e)}{4 b^2 x (b+c x)^2}-\frac{3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac{3 \sqrt{d+e x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2 (b+c x)}+\frac{(d+e x)^{3/2} (c d-b e) \left (2 b^2 e^2-17 b c d e+12 c^2 d^2\right )}{4 b^3 c (b+c x)^2}-\frac{d (d+e x)^{7/2}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

(3*(c*d - b*e)*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4
*b^4*c^2*(b + c*x)) + ((c*d - b*e)*(12*c^2*d^2 - 17*b*c*d*e + 2*b^2*e^2)*(d + e*
x)^(3/2))/(4*b^3*c*(b + c*x)^2) + (d*(8*c*d - 11*b*e)*(d + e*x)^(5/2))/(4*b^2*x*
(b + c*x)^2) - (d*(d + e*x)^(7/2))/(2*b*x^2*(b + c*x)^2) - (3*d^(5/2)*(16*c^2*d^
2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d -
b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/S
qrt[c*d - b*e]])/(4*b^5*c^(5/2))

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Rubi in Sympy [A]  time = 126.67, size = 296, normalized size = 0.95 \[ - \frac{d \left (d + e x\right )^{\frac{7}{2}}}{2 b x^{2} \left (b + c x\right )^{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{2 b^{2} c x \left (b + c x\right )^{2}} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right ) \left (3 b^{2} e^{2} + 7 b c d e - 12 c^{2} d^{2}\right )}{4 b^{3} c^{2} x \left (b + c x\right )} + \frac{3 d \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b^{2} e^{2} + 4 b c d e - 4 c^{2} d^{2}\right )}{4 b^{4} c^{2} x} - \frac{3 d^{\frac{5}{2}} \left (21 b^{2} e^{2} - 36 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{4 b^{5}} + \frac{3 \left (b e - c d\right )^{\frac{5}{2}} \left (b^{2} e^{2} + 4 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{4 b^{5} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(9/2)/(c*x**2+b*x)**3,x)

[Out]

-d*(d + e*x)**(7/2)/(2*b*x**2*(b + c*x)**2) - (d + e*x)**(5/2)*(b*e - 2*c*d)*(b*
e - c*d)/(2*b**2*c*x*(b + c*x)**2) - (d + e*x)**(3/2)*(b*e - c*d)*(3*b**2*e**2 +
7*b*c*d*e - 12*c**2*d**2)/(4*b**3*c**2*x*(b + c*x)) + 3*d*sqrt(d + e*x)*(b*e -
2*c*d)*(b**2*e**2 + 4*b*c*d*e - 4*c**2*d**2)/(4*b**4*c**2*x) - 3*d**(5/2)*(21*b*
*2*e**2 - 36*b*c*d*e + 16*c**2*d**2)*atanh(sqrt(d + e*x)/sqrt(d))/(4*b**5) + 3*(
b*e - c*d)**(5/2)*(b**2*e**2 + 4*b*c*d*e + 16*c**2*d**2)*atan(sqrt(c)*sqrt(d + e
*x)/sqrt(b*e - c*d))/(4*b**5*c**(5/2))

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Mathematica [A]  time = 0.649088, size = 213, normalized size = 0.68 \[ \frac{-3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{5/2}}+b \sqrt{d+e x} \left (\frac{(c d-b e)^3 (5 b e+12 c d)}{c^2 (b+c x)}+\frac{2 b (c d-b e)^4}{c^2 (b+c x)^2}+\frac{d^3 (12 c d-17 b e)}{x}-\frac{2 b d^4}{x^2}\right )}{4 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

(b*Sqrt[d + e*x]*((-2*b*d^4)/x^2 + (d^3*(12*c*d - 17*b*e))/x + (2*b*(c*d - b*e)^
4)/(c^2*(b + c*x)^2) + ((c*d - b*e)^3*(12*c*d + 5*b*e))/(c^2*(b + c*x))) - 3*d^(
5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (3*
(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[c*d - b*e]])/c^(5/2))/(4*b^5)

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Maple [B]  time = 0.041, size = 703, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(c*x^2+b*x)^3,x)

[Out]

45/4*e^2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(1/2)*d^4-3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(
1/2)*d^5*c^3+3/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c
)^(1/2))*d+21/4*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)
^(1/2))*d^2-111/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2))*d^3+33*e/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*
d)*c)^(1/2))*d^4+3/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d+21/4*e^3/b^2/(c*e*x+b*e
)^2*c*(e*x+d)^(3/2)*d^2-31/4*e^2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(3/2)*d^3+3*e/b^4
/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d^4*c^3+15/2*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-
15*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*d^3-5/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(3/
2)-3/4*e^6*b/(c*e*x+b*e)^2/c^2*(e*x+d)^(1/2)+3/4*e^5/c^2/((b*e-c*d)*c)^(1/2)*arc
tan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-12/b^5/((b*e-c*d)*c)^(1/2)*arctan(c*(e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^5*c^3-17/4*d^3/b^3/x^2*(e*x+d)^(3/2)+3/e*d^4/b
^4/x^2*(e*x+d)^(3/2)*c+15/4*d^4/b^3/x^2*(e*x+d)^(1/2)-3/e*d^5/b^4/x^2*(e*x+d)^(1
/2)*c-63/4*e^2*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))+27*e*d^(7/2)/b^4*arcta
nh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(9/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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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((e*x + d)^(9/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76463, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^
4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*
c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*
e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e
+ 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((16*c^6*d^4 - 36*b*c^5
*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c
^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*
sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*
c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)
*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 -
3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*
x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2
*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4
*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4
- 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt(-(c*
d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + 3*((16*c^6*d^4 - 36*b*c
^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3
*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2
)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*
b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^
4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3
- 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^
4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), -1/8*(6*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21
*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)
*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(-d)*ar
ctan(sqrt(e*x + d)/sqrt(-d)) - 3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e
^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9
*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c
^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt((c*d - b*e)/c)
*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) +
2*(2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4
*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*
d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)
*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), -1/4*(3*((16*c^6*d^
4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*
e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^
2*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - 3*((16*c^6*d^4 - 28*b*c^5*
d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4
- 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16
*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x
^2)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (2*b^4*c^2
*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 -
5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*
b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x +
d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]

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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((e*x+d)**(9/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242984, size = 852, normalized size = 2.73 \[ \frac{3 \,{\left (16 \, c^{2} d^{5} - 36 \, b c d^{4} e + 21 \, b^{2} d^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} - \frac{3 \,{\left (16 \, c^{5} d^{5} - 44 \, b c^{4} d^{4} e + 37 \, b^{2} c^{3} d^{3} e^{2} - 7 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4} - b^{5} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5} c^{2}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{5} d^{4} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{5} d^{5} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{6} e - 24 \, \sqrt{x e + d} c^{5} d^{7} e - 48 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{4} d^{3} e^{2} + 180 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{4} d^{4} e^{2} - 216 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{5} e^{2} + 84 \, \sqrt{x e + d} b c^{4} d^{6} e^{2} + 21 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{3} d^{2} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{3} d^{3} e^{3} + 217 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{4} e^{3} - 102 \, \sqrt{x e + d} b^{2} c^{3} d^{5} e^{3} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} c^{2} d e^{4} + 24 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c^{2} d^{2} e^{4} - 74 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d^{3} e^{4} + 45 \, \sqrt{x e + d} b^{3} c^{2} d^{4} e^{4} - 5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} c e^{5} + 10 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} c d e^{5} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c d^{2} e^{5} - 3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{6} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{6} - 3 \, \sqrt{x e + d} b^{5} d^{2} e^{6}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

3/4*(16*c^2*d^5 - 36*b*c*d^4*e + 21*b^2*d^3*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/
(b^5*sqrt(-d)) - 3/4*(16*c^5*d^5 - 44*b*c^4*d^4*e + 37*b^2*c^3*d^3*e^2 - 7*b^3*c
^2*d^2*e^3 - b^4*c*d*e^4 - b^5*e^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))
/(sqrt(-c^2*d + b*c*e)*b^5*c^2) + 1/4*(24*(x*e + d)^(7/2)*c^5*d^4*e - 72*(x*e +
d)^(5/2)*c^5*d^5*e + 72*(x*e + d)^(3/2)*c^5*d^6*e - 24*sqrt(x*e + d)*c^5*d^7*e -
48*(x*e + d)^(7/2)*b*c^4*d^3*e^2 + 180*(x*e + d)^(5/2)*b*c^4*d^4*e^2 - 216*(x*e
+ d)^(3/2)*b*c^4*d^5*e^2 + 84*sqrt(x*e + d)*b*c^4*d^6*e^2 + 21*(x*e + d)^(7/2)*
b^2*c^3*d^2*e^3 - 136*(x*e + d)^(5/2)*b^2*c^3*d^3*e^3 + 217*(x*e + d)^(3/2)*b^2*
c^3*d^4*e^3 - 102*sqrt(x*e + d)*b^2*c^3*d^5*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*d*e^
4 + 24*(x*e + d)^(5/2)*b^3*c^2*d^2*e^4 - 74*(x*e + d)^(3/2)*b^3*c^2*d^3*e^4 + 45
*sqrt(x*e + d)*b^3*c^2*d^4*e^4 - 5*(x*e + d)^(7/2)*b^4*c*e^5 + 10*(x*e + d)^(5/2
)*b^4*c*d*e^5 - 5*(x*e + d)^(3/2)*b^4*c*d^2*e^5 - 3*(x*e + d)^(5/2)*b^5*e^6 + 6*
(x*e + d)^(3/2)*b^5*d*e^6 - 3*sqrt(x*e + d)*b^5*d^2*e^6)/(((x*e + d)^2*c - 2*(x*
e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^2)