3.1978 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=238 \[ \frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5 (a+b x) (d+e x)^2}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^5 (a+b x)}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)}-\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (3 b d-4 a e)}{e^4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.385497, antiderivative size = 238, 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 \[ \frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5 (a+b x) (d+e x)^2}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^5 (a+b x)}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)}-\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (3 b d-4 a e)}{e^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.8881, size = 187, normalized size = 0.79 \[ \frac{b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{3}} + \frac{6 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{4}} + \frac{6 b^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} - \frac{2 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{2} \left (d + e x\right )} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{2}} \]

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)**(3/2)/(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.249891, size = 185, normalized size = 0.78 \[ \frac{\sqrt{(a+b x)^2} \left (-a^4 e^4-4 a^3 b e^3 (d+2 e x)+6 a^2 b^2 d e^2 (3 d+4 e x)+4 a b^3 e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+12 b^2 (d+e x)^2 (b d-a e)^2 \log (d+e x)+b^4 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )}{2 e^5 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(-(a^4*e^4) - 4*a^3*b*e^3*(d + 2*e*x) + 6*a^2*b^2*d*e^2*(3*d
+ 4*e*x) + 4*a*b^3*e*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + b^4*(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) + 12*b^2*(b*d - a*e)^2*(d
+ e*x)^2*Log[d + e*x]))/(2*e^5*(a + b*x)*(d + e*x)^2)

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Maple [A]  time = 0.025, size = 350, normalized size = 1.5 \[{\frac{{x}^{4}{b}^{4}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+8\,{x}^{3}a{b}^{3}{e}^{4}-4\,{x}^{3}{b}^{4}d{e}^{3}+24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-48\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+24\,\ln \left ( ex+d \right ) x{b}^{4}{d}^{3}e+16\,{x}^{2}a{b}^{3}d{e}^{3}-11\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,\ln \left ( ex+d \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}-8\,x{a}^{3}b{e}^{4}+24\,x{a}^{2}{b}^{2}d{e}^{3}-16\,xa{b}^{3}{d}^{2}{e}^{2}+2\,x{b}^{4}{d}^{3}e-{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-20\,a{b}^{3}{d}^{3}e+7\,{b}^{4}{d}^{4}}{2\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^3,x)

[Out]

1/2*((b*x+a)^2)^(3/2)*(x^4*b^4*e^4+12*ln(e*x+d)*x^2*a^2*b^2*e^4-24*ln(e*x+d)*x^2
*a*b^3*d*e^3+12*ln(e*x+d)*x^2*b^4*d^2*e^2+8*x^3*a*b^3*e^4-4*x^3*b^4*d*e^3+24*ln(
e*x+d)*x*a^2*b^2*d*e^3-48*ln(e*x+d)*x*a*b^3*d^2*e^2+24*ln(e*x+d)*x*b^4*d^3*e+16*
x^2*a*b^3*d*e^3-11*x^2*b^4*d^2*e^2+12*ln(e*x+d)*a^2*b^2*d^2*e^2-24*ln(e*x+d)*a*b
^3*d^3*e+12*ln(e*x+d)*b^4*d^4-8*x*a^3*b*e^4+24*x*a^2*b^2*d*e^3-16*x*a*b^3*d^2*e^
2+2*x*b^4*d^3*e-a^4*e^4-4*a^3*b*d*e^3+18*a^2*b^2*d^2*e^2-20*a*b^3*d^3*e+7*b^4*d^
4)/(b*x+a)^3/e^5/(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)^(3/2)*(b*x + a)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287095, size = 393, normalized size = 1.65 \[ \frac{b^{4} e^{4} x^{4} + 7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} - 4 \,{\left (b^{4} d e^{3} - 2 \, a b^{3} e^{4}\right )} x^{3} -{\left (11 \, b^{4} d^{2} e^{2} - 16 \, a b^{3} d e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 12 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 2 \, a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^4*e^4*x^4 + 7*b^4*d^4 - 20*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e
^3 - a^4*e^4 - 4*(b^4*d*e^3 - 2*a*b^3*e^4)*x^3 - (11*b^4*d^2*e^2 - 16*a*b^3*d*e^
3)*x^2 + 2*(b^4*d^3*e - 8*a*b^3*d^2*e^2 + 12*a^2*b^2*d*e^3 - 4*a^3*b*e^4)*x + 12
*(b^4*d^4 - 2*a*b^3*d^3*e + a^2*b^2*d^2*e^2 + (b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2
*b^2*e^4)*x^2 + 2*(b^4*d^3*e - 2*a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(e*x + d))
/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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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)**(3/2)/(e*x+d)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.309905, size = 358, normalized size = 1.5 \[ 6 \,{\left (b^{4} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b^{3} d e{\rm sign}\left (b x + a\right ) + a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{4} x^{2} e^{3}{\rm sign}\left (b x + a\right ) - 6 \, b^{4} d x e^{2}{\rm sign}\left (b x + a\right ) + 8 \, a b^{3} x e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )} + \frac{{\left (7 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 20 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 18 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) - a^{4} e^{4}{\rm sign}\left (b x + a\right ) + 8 \,{\left (b^{4} d^{3} e{\rm sign}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - a^{3} b e^{4}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(b^4*d^2*sign(b*x + a) - 2*a*b^3*d*e*sign(b*x + a) + a^2*b^2*e^2*sign(b*x + a)
)*e^(-5)*ln(abs(x*e + d)) + 1/2*(b^4*x^2*e^3*sign(b*x + a) - 6*b^4*d*x*e^2*sign(
b*x + a) + 8*a*b^3*x*e^3*sign(b*x + a))*e^(-6) + 1/2*(7*b^4*d^4*sign(b*x + a) -
20*a*b^3*d^3*e*sign(b*x + a) + 18*a^2*b^2*d^2*e^2*sign(b*x + a) - 4*a^3*b*d*e^3*
sign(b*x + a) - a^4*e^4*sign(b*x + a) + 8*(b^4*d^3*e*sign(b*x + a) - 3*a*b^3*d^2
*e^2*sign(b*x + a) + 3*a^2*b^2*d*e^3*sign(b*x + a) - a^3*b*e^4*sign(b*x + a))*x)
*e^(-5)/(x*e + d)^2