3.2002 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=344 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac{b^6 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)} \]

[Out]

-((b^5*(5*b*d - 6*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^6*
x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^4) + (2*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)^3) - (15*b^2*(b*d - a*e)^4*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (20*b^3*(b*d - a*e)^3*
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
*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 = 0.585766, antiderivative size = 344, 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 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac{b^6 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((b^5*(5*b*d - 6*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^6*
x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^4) + (2*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)^3) - (15*b^2*(b*d - a*e)^4*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (20*b^3*(b*d - a*e)^3*
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
*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 = 54.4598, size = 270, normalized size = 0.78 \[ \frac{5 b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e^{5}} + \frac{15 b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{5 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4} \left (d + e x\right )} - \frac{b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e^{3} \left (d + e x\right )^{2}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{2 e^{2} \left (d + e x\right )^{3}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{4 e \left (d + e x\right )^{4}} \]

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)**5,x)

[Out]

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

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Mathematica [A]  time = 0.311291, size = 318, normalized size = 0.92 \[ -\frac{\sqrt{(a+b x)^2} \left (a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-57 d^6-168 d^5 e x-132 d^4 e^2 x^2+32 d^3 e^3 x^3+68 d^2 e^4 x^4+12 d e^5 x^5-2 e^6 x^6\right )\right )}{4 e^7 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*
d*e*x + 6*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)
- 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 2*a*b^5*e
*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^
5*x^5) + b^6*(-57*d^6 - 168*d^5*e*x - 132*d^4*e^2*x^2 + 32*d^3*e^3*x^3 + 68*d^2*
e^4*x^4 + 12*d*e^5*x^5 - 2*e^6*x^6) - 60*b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e
*x]))/(4*e^7*(a + b*x)*(d + e*x)^4)

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Maple [B]  time = 0.028, size = 670, normalized size = 2. \[{\frac{240\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+60\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}-120\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+540\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-504\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-80\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-120\,\ln \left ( ex+d \right ){x}^{4}a{b}^{5}d{e}^{5}+168\,x{b}^{6}{d}^{5}e-68\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-{a}^{6}{e}^{6}+57\,{b}^{6}{d}^{6}+60\,\ln \left ( ex+d \right ){x}^{4}{a}^{2}{b}^{4}{e}^{6}-154\,{d}^{5}a{b}^{5}e-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+2\,{x}^{6}{b}^{6}{e}^{6}+240\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}-120\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+96\,{x}^{4}a{b}^{5}d{e}^{5}+360\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-720\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+240\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-480\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}-480\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}+240\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+125\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+440\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-496\,xa{b}^{5}{d}^{4}{e}^{2}-96\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-20\,x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+360\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}-5\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}-2\,{a}^{5}bd{e}^{5}+24\,{x}^{5}a{b}^{5}{e}^{6}-12\,{x}^{5}{b}^{6}d{e}^{5}-32\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-80\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-30\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+132\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-8\,x{a}^{5}b{e}^{6}}{4\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^5,x)

[Out]

1/4*((b*x+a)^2)^(5/2)*(240*x^3*a^2*b^4*d*e^5+60*ln(e*x+d)*a^2*b^4*d^4*e^2-120*ln
(e*x+d)*a*b^5*d^5*e+540*x^2*a^2*b^4*d^2*e^4-504*x^2*a*b^5*d^3*e^3-80*x*a^3*b^3*d
^2*e^4-120*ln(e*x+d)*x^4*a*b^5*d*e^5+168*x*b^6*d^5*e-68*x^4*b^6*d^2*e^4-a^6*e^6+
57*b^6*d^6+60*ln(e*x+d)*x^4*a^2*b^4*e^6-154*d^5*a*b^5*e-20*a^3*b^3*d^3*e^3+60*ln
(e*x+d)*b^6*d^6+2*x^6*b^6*e^6+240*ln(e*x+d)*x^3*b^6*d^3*e^3-120*x^2*a^3*b^3*d*e^
5+96*x^4*a*b^5*d*e^5+360*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4-720*ln(e*x+d)*x^2*a*b^5*d
^3*e^3+240*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^2-480*ln(e*x+
d)*x^3*a*b^5*d^2*e^4+240*ln(e*x+d)*x^3*a^2*b^4*d*e^5+125*d^4*e^2*a^2*b^4+440*x*a
^2*b^4*d^3*e^3-496*x*a*b^5*d^4*e^2-96*x^3*a*b^5*d^2*e^4-20*x*a^4*b^2*d*e^5+240*l
n(e*x+d)*x*b^6*d^5*e+360*ln(e*x+d)*x^2*b^6*d^4*e^2-5*b^2*a^4*d^2*e^4+60*ln(e*x+d
)*x^4*b^6*d^2*e^4-2*a^5*b*d*e^5+24*x^5*a*b^5*e^6-12*x^5*b^6*d*e^5-32*x^3*b^6*d^3
*e^3-80*x^3*a^3*b^3*e^6-30*x^2*a^4*b^2*e^6+132*x^2*b^6*d^4*e^2-8*x*a^5*b*e^6)/(b
*x+a)^5/e^7/(e*x+d)^4

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289181, size = 771, normalized size = 2.24 \[ \frac{2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \,{\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \,{\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \,{\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \,{\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3
*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6*d*e^5 - 2*a
*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16*(2*b^6*d^3*e^3 + 6*
a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a
*b^5*d^3*e^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 4*(4
2*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a
^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*d^4*e^2 +
(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e
^4 + a^2*b^4*d*e^5)*x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^
2 + 4*(b^6*d^5*e - 2*a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3)*x)*log(e*x + d))/(e^11*x^4
+ 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.30079, size = 680, normalized size = 1.98 \[ 15 \,{\left (b^{6} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b^{5} d e{\rm sign}\left (b x + a\right ) + a^{2} b^{4} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{6} x^{2} e^{5}{\rm sign}\left (b x + a\right ) - 10 \, b^{6} d x e^{4}{\rm sign}\left (b x + a\right ) + 12 \, a b^{5} x e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-10\right )} + \frac{{\left (57 \, b^{6} d^{6}{\rm sign}\left (b x + a\right ) - 154 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 125 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 5 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 2 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) - a^{6} e^{6}{\rm sign}\left (b x + a\right ) + 80 \,{\left (b^{6} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5}{\rm sign}\left (b x + a\right ) - a^{3} b^{3} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{3} + 30 \,{\left (7 \, b^{6} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 20 \, a b^{5} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 18 \, a^{2} b^{4} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5}{\rm sign}\left (b x + a\right ) - a^{4} b^{2} e^{6}{\rm sign}\left (b x + a\right )\right )} x^{2} + 4 \,{\left (47 \, b^{6} d^{5} e{\rm sign}\left (b x + a\right ) - 130 \, a b^{5} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 110 \, a^{2} b^{4} d^{3} e^{3}{\rm sign}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 5 \, a^{4} b^{2} d e^{5}{\rm sign}\left (b x + a\right ) - 2 \, a^{5} b e^{6}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{4 \,{\left (x e + d\right )}^{4}} \]

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)^5,x, algorithm="giac")

[Out]

15*(b^6*d^2*sign(b*x + a) - 2*a*b^5*d*e*sign(b*x + a) + a^2*b^4*e^2*sign(b*x + a
))*e^(-7)*ln(abs(x*e + d)) + 1/2*(b^6*x^2*e^5*sign(b*x + a) - 10*b^6*d*x*e^4*sig
n(b*x + a) + 12*a*b^5*x*e^5*sign(b*x + a))*e^(-10) + 1/4*(57*b^6*d^6*sign(b*x +
a) - 154*a*b^5*d^5*e*sign(b*x + a) + 125*a^2*b^4*d^4*e^2*sign(b*x + a) - 20*a^3*
b^3*d^3*e^3*sign(b*x + a) - 5*a^4*b^2*d^2*e^4*sign(b*x + a) - 2*a^5*b*d*e^5*sign
(b*x + a) - a^6*e^6*sign(b*x + a) + 80*(b^6*d^3*e^3*sign(b*x + a) - 3*a*b^5*d^2*
e^4*sign(b*x + a) + 3*a^2*b^4*d*e^5*sign(b*x + a) - a^3*b^3*e^6*sign(b*x + a))*x
^3 + 30*(7*b^6*d^4*e^2*sign(b*x + a) - 20*a*b^5*d^3*e^3*sign(b*x + a) + 18*a^2*b
^4*d^2*e^4*sign(b*x + a) - 4*a^3*b^3*d*e^5*sign(b*x + a) - a^4*b^2*e^6*sign(b*x
+ a))*x^2 + 4*(47*b^6*d^5*e*sign(b*x + a) - 130*a*b^5*d^4*e^2*sign(b*x + a) + 11
0*a^2*b^4*d^3*e^3*sign(b*x + a) - 20*a^3*b^3*d^2*e^4*sign(b*x + a) - 5*a^4*b^2*d
*e^5*sign(b*x + a) - 2*a^5*b*e^6*sign(b*x + a))*x)*e^(-7)/(x*e + d)^4