3.2000 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=347 \[ \frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac{20 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)} \]

[Out]

(-20*b^3*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d
- a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b*(b*
d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (15*b^4*(b
*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) - (2*b^
5*(b*d - a*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*
(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) + (15*b^2*(b*d - a*
e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi [A]  time = 0.624554, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac{20 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-20*b^3*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d
- a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b*(b*
d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (15*b^4*(b
*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) - (2*b^
5*(b*d - a*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*
(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) + (15*b^2*(b*d - a*
e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi in Sympy [A]  time = 60.2471, size = 272, normalized size = 0.78 \[ \frac{15 b^{2} \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e^{3}} + \frac{5 b^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4}} + \frac{5 b^{2} \left (3 a + 3 b x\right ) \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e^{5}} + \frac{15 b^{2} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{15 b^{2} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{3 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{2 e \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**3,x)

[Out]

15*b**2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(4*e**3) + 5*b**2*(a*e - b
*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/e**4 + 5*b**2*(3*a + 3*b*x)*(a*e - b*d)*
*2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2*e**5) + 15*b**2*(a*e - b*d)**3*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/e**6 + 15*b**2*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*
x**2)*log(d + e*x)/(e**7*(a + b*x)) - 3*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e
**2*(d + e*x)) - (a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(2*e*(d + e*x)**2
)

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Mathematica [A]  time = 0.298981, size = 321, normalized size = 0.93 \[ \frac{\sqrt{(a+b x)^2} \left (-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )\right )}{4 e^7 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(Sqrt[(a + b*x)^2]*(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*
d + 4*e*x) + 40*a^3*b^3*e^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 30*
a^2*b^4*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*a*b
^5*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^
5*x^5) + b^6*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*
x^4 - 2*d*e^5*x^5 + e^6*x^6) + 60*b^2*(b*d - a*e)^4*(d + e*x)^2*Log[d + e*x]))/(
4*e^7*(a + b*x)*(d + e*x)^2)

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Maple [B]  time = 0.029, size = 669, normalized size = 1.9 \[{\frac{-120\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-240\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}+360\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}-240\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+60\,\ln \left ( ex+d \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}-330\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+252\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-16\,x{b}^{6}{d}^{5}e+5\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-2\,{a}^{6}{e}^{6}+22\,{b}^{6}{d}^{6}-108\,{d}^{5}a{b}^{5}e-200\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+{x}^{6}{b}^{6}{e}^{6}+160\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-20\,{x}^{4}a{b}^{5}d{e}^{5}-240\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}+360\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-240\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+120\,\ln \left ( ex+d \right ) x{a}^{4}{b}^{2}d{e}^{5}-480\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+720\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-480\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}+210\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+60\,\ln \left ( ex+d \right ){x}^{2}{a}^{4}{b}^{2}{e}^{6}+60\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+24\,xa{b}^{5}{d}^{4}{e}^{2}+80\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+120\,x{a}^{4}{b}^{2}d{e}^{5}+120\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+60\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}+90\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-12\,{a}^{5}bd{e}^{5}+8\,{x}^{5}a{b}^{5}{e}^{6}-2\,{x}^{5}{b}^{6}d{e}^{5}-20\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+80\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+30\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-68\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-24\,x{a}^{5}b{e}^{6}}{4\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x)

[Out]

1/4*((b*x+a)^2)^(5/2)*(-120*x^3*a^2*b^4*d*e^5-240*ln(e*x+d)*a^3*b^3*d^3*e^3+360*
ln(e*x+d)*a^2*b^4*d^4*e^2-240*ln(e*x+d)*a*b^5*d^5*e+60*ln(e*x+d)*a^4*b^2*d^2*e^4
-330*x^2*a^2*b^4*d^2*e^4+252*x^2*a*b^5*d^3*e^3-160*x*a^3*b^3*d^2*e^4-16*x*b^6*d^
5*e+5*x^4*b^6*d^2*e^4-2*a^6*e^6+22*b^6*d^6-108*d^5*a*b^5*e-200*a^3*b^3*d^3*e^3+6
0*ln(e*x+d)*b^6*d^6+x^6*b^6*e^6+160*x^2*a^3*b^3*d*e^5-20*x^4*a*b^5*d*e^5-240*ln(
e*x+d)*x^2*a^3*b^3*d*e^5+360*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4-240*ln(e*x+d)*x^2*a*b
^5*d^3*e^3+120*ln(e*x+d)*x*a^4*b^2*d*e^5-480*ln(e*x+d)*x*a^3*b^3*d^2*e^4+720*ln(
e*x+d)*x*a^2*b^4*d^3*e^3-480*ln(e*x+d)*x*a*b^5*d^4*e^2+210*d^4*e^2*a^2*b^4+60*ln
(e*x+d)*x^2*a^4*b^2*e^6+60*x*a^2*b^4*d^3*e^3+24*x*a*b^5*d^4*e^2+80*x^3*a*b^5*d^2
*e^4+120*x*a^4*b^2*d*e^5+120*ln(e*x+d)*x*b^6*d^5*e+60*ln(e*x+d)*x^2*b^6*d^4*e^2+
90*b^2*a^4*d^2*e^4-12*a^5*b*d*e^5+8*x^5*a*b^5*e^6-2*x^5*b^6*d*e^5-20*x^3*b^6*d^3
*e^3+80*x^3*a^3*b^3*e^6+30*x^4*a^2*b^4*e^6-68*x^2*b^6*d^4*e^2-24*x*a^5*b*e^6)/(b
*x+a)^5/e^7/(e*x+d)^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((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285006, size = 740, normalized size = 2.13 \[ \frac{b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \,{\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 2 \,{\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 200*a^3*
b^3*d^3*e^3 + 90*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b^6*d*e^5 - 4
*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 - 20*(b^6*
d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3*e^6)*x^3 - 2*(34*b^6*d^4
*e^2 - 126*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 80*a^3*b^3*d*e^5)*x^2 - 4*(4*b^
6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d^3*e^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2
*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 - 4*a^
3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2
*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a
^2*b^4*d^3*e^3 - 4*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(e^9*x^2 +
2*d*e^8*x + d^2*e^7)

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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((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287166, size = 687, normalized size = 1.98 \[ 15 \,{\left (b^{6} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} b^{2} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (b^{6} x^{4} e^{9}{\rm sign}\left (b x + a\right ) - 4 \, b^{6} d x^{3} e^{8}{\rm sign}\left (b x + a\right ) + 12 \, b^{6} d^{2} x^{2} e^{7}{\rm sign}\left (b x + a\right ) - 40 \, b^{6} d^{3} x e^{6}{\rm sign}\left (b x + a\right ) + 8 \, a b^{5} x^{3} e^{9}{\rm sign}\left (b x + a\right ) - 36 \, a b^{5} d x^{2} e^{8}{\rm sign}\left (b x + a\right ) + 144 \, a b^{5} d^{2} x e^{7}{\rm sign}\left (b x + a\right ) + 30 \, a^{2} b^{4} x^{2} e^{9}{\rm sign}\left (b x + a\right ) - 180 \, a^{2} b^{4} d x e^{8}{\rm sign}\left (b x + a\right ) + 80 \, a^{3} b^{3} x e^{9}{\rm sign}\left (b x + a\right )\right )} e^{\left (-12\right )} + \frac{{\left (11 \, b^{6} d^{6}{\rm sign}\left (b x + a\right ) - 54 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 105 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 100 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 45 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 6 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) - a^{6} e^{6}{\rm sign}\left (b x + a\right ) + 12 \,{\left (b^{6} d^{5} e{\rm sign}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5}{\rm sign}\left (b x + a\right ) - a^{5} b e^{6}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^3,x, algorithm="giac")

[Out]

15*(b^6*d^4*sign(b*x + a) - 4*a*b^5*d^3*e*sign(b*x + a) + 6*a^2*b^4*d^2*e^2*sign
(b*x + a) - 4*a^3*b^3*d*e^3*sign(b*x + a) + a^4*b^2*e^4*sign(b*x + a))*e^(-7)*ln
(abs(x*e + d)) + 1/4*(b^6*x^4*e^9*sign(b*x + a) - 4*b^6*d*x^3*e^8*sign(b*x + a)
+ 12*b^6*d^2*x^2*e^7*sign(b*x + a) - 40*b^6*d^3*x*e^6*sign(b*x + a) + 8*a*b^5*x^
3*e^9*sign(b*x + a) - 36*a*b^5*d*x^2*e^8*sign(b*x + a) + 144*a*b^5*d^2*x*e^7*sig
n(b*x + a) + 30*a^2*b^4*x^2*e^9*sign(b*x + a) - 180*a^2*b^4*d*x*e^8*sign(b*x + a
) + 80*a^3*b^3*x*e^9*sign(b*x + a))*e^(-12) + 1/2*(11*b^6*d^6*sign(b*x + a) - 54
*a*b^5*d^5*e*sign(b*x + a) + 105*a^2*b^4*d^4*e^2*sign(b*x + a) - 100*a^3*b^3*d^3
*e^3*sign(b*x + a) + 45*a^4*b^2*d^2*e^4*sign(b*x + a) - 6*a^5*b*d*e^5*sign(b*x +
a) - a^6*e^6*sign(b*x + a) + 12*(b^6*d^5*e*sign(b*x + a) - 5*a*b^5*d^4*e^2*sign
(b*x + a) + 10*a^2*b^4*d^3*e^3*sign(b*x + a) - 10*a^3*b^3*d^2*e^4*sign(b*x + a)
+ 5*a^4*b^2*d*e^5*sign(b*x + a) - a^5*b*e^6*sign(b*x + a))*x)*e^(-7)/(x*e + d)^2