3.1976 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=210 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e} \]

[Out]

-((b*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x))) + ((b*d - a
*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^3) - ((b*d - a*e)*(a + b*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^2) + ((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(4*e) + ((b*d - a*e)^4*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 = 0.288456, antiderivative size = 210, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x))) + ((b*d - a
*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^3) - ((b*d - a*e)*(a + b*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^2) + ((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(4*e) + ((b*d - a*e)^4*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 = 42.4341, size = 178, normalized size = 0.85 \[ \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e} + \frac{\left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{\left (3 a + 3 b x\right ) \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{3}} + \frac{\left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{4}} + \frac{\left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.131875, size = 133, normalized size = 0.63 \[ \frac{\sqrt{(a+b x)^2} \left (b e x \left (48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)\right )}{12 e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(b*e*x*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6
*d^2 - 3*d*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3
)) + 12*(b*d - a*e)^4*Log[d + e*x]))/(12*e^5*(a + b*x))

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Maple [A]  time = 0.016, size = 225, normalized size = 1.1 \[{\frac{3\,{x}^{4}{b}^{4}{e}^{4}+16\,{x}^{3}a{b}^{3}{e}^{4}-4\,{x}^{3}{b}^{4}d{e}^{3}+36\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,{x}^{2}a{b}^{3}d{e}^{3}+6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{4}{e}^{4}-48\,\ln \left ( ex+d \right ){a}^{3}bd{e}^{3}+72\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-48\,\ln \left ( ex+d \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}+48\,x{a}^{3}b{e}^{4}-72\,x{a}^{2}{b}^{2}d{e}^{3}+48\,xa{b}^{3}{d}^{2}{e}^{2}-12\,x{b}^{4}{d}^{3}e}{12\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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),x)

[Out]

1/12*((b*x+a)^2)^(3/2)*(3*x^4*b^4*e^4+16*x^3*a*b^3*e^4-4*x^3*b^4*d*e^3+36*x^2*a^
2*b^2*e^4-24*x^2*a*b^3*d*e^3+6*x^2*b^4*d^2*e^2+12*ln(e*x+d)*a^4*e^4-48*ln(e*x+d)
*a^3*b*d*e^3+72*ln(e*x+d)*a^2*b^2*d^2*e^2-48*ln(e*x+d)*a*b^3*d^3*e+12*ln(e*x+d)*
b^4*d^4+48*x*a^3*b*e^4-72*x*a^2*b^2*d*e^3+48*x*a*b^3*d^2*e^2-12*x*b^4*d^3*e)/(b*
x+a)^3/e^5

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278075, size = 242, normalized size = 1.15 \[ \frac{3 \, b^{4} e^{4} x^{4} - 4 \,{\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]

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),x, algorithm="fricas")

[Out]

1/12*(3*b^4*e^4*x^4 - 4*(b^4*d*e^3 - 4*a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 - 4*a*b^3
*d*e^3 + 6*a^2*b^2*e^4)*x^2 - 12*(b^4*d^3*e - 4*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3
- 4*a^3*b*e^4)*x + 12*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e
^3 + a^4*e^4)*log(e*x + d))/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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287033, size = 359, normalized size = 1.71 \[{\left (b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{4} x^{4} e^{3}{\rm sign}\left (b x + a\right ) - 4 \, b^{4} d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{4} d^{2} x^{2} e{\rm sign}\left (b x + a\right ) - 12 \, b^{4} d^{3} x{\rm sign}\left (b x + a\right ) + 16 \, a b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) - 24 \, a b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 48 \, a b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + 36 \, a^{2} b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) - 72 \, a^{2} b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + 48 \, a^{3} b x e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )} \]

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),x, algorithm="giac")

[Out]

(b^4*d^4*sign(b*x + a) - 4*a*b^3*d^3*e*sign(b*x + a) + 6*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))*e^(-5)*ln(abs(x*e
+ d)) + 1/12*(3*b^4*x^4*e^3*sign(b*x + a) - 4*b^4*d*x^3*e^2*sign(b*x + a) + 6*b^
4*d^2*x^2*e*sign(b*x + a) - 12*b^4*d^3*x*sign(b*x + a) + 16*a*b^3*x^3*e^3*sign(b
*x + a) - 24*a*b^3*d*x^2*e^2*sign(b*x + a) + 48*a*b^3*d^2*x*e*sign(b*x + a) + 36
*a^2*b^2*x^2*e^3*sign(b*x + a) - 72*a^2*b^2*d*x*e^2*sign(b*x + a) + 48*a^3*b*x*e
^3*sign(b*x + a))*e^(-4)