3.1980 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=246 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^2}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^5 (a+b x) (d+e x)^4}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)} \]

[Out]

-((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^5*(a + b*x)*(d + e*x)^4) + (
4*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^3) -
(3*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*(d + e*x)^2)
+ (4*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*(d + e*x)) +
(b^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 = 0.403655, antiderivative size = 246, 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 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^2}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^5 (a+b x) (d+e x)^4}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^5*(a + b*x)*(d + e*x)^4) + (
4*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^3) -
(3*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*(d + e*x)^2)
+ (4*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*(d + e*x)) +
(b^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 = 47.6921, size = 196, normalized size = 0.8 \[ \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} - \frac{b^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5} \left (a + b x\right ) \left (d + e x\right )} - \frac{b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{3} \left (d + e x\right )^{2}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{3}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e \left (d + e x\right )^{4}} \]

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

[Out]

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

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Mathematica [A]  time = 0.200329, size = 144, normalized size = 0.59 \[ \frac{\sqrt{(a+b x)^2} \left ((b d-a e) \left (3 a^3 e^3+a^2 b e^2 (7 d+16 e x)+a b^2 e \left (13 d^2+40 d e x+36 e^2 x^2\right )+b^3 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 b^4 (d+e x)^4 \log (d+e x)\right )}{12 e^5 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*((b*d - a*e)*(3*a^3*e^3 + a^2*b*e^2*(7*d + 16*e*x) + a*b^2*e*
(13*d^2 + 40*d*e*x + 36*e^2*x^2) + b^3*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48
*e^3*x^3)) + 12*b^4*(d + e*x)^4*Log[d + e*x]))/(12*e^5*(a + b*x)*(d + e*x)^4)

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Maple [A]  time = 0.02, size = 276, normalized size = 1.1 \[{\frac{12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}+48\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+72\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-48\,{x}^{3}a{b}^{3}{e}^{4}+48\,{x}^{3}{b}^{4}d{e}^{3}+48\,\ln \left ( ex+d \right ) x{b}^{4}{d}^{3}e-36\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-72\,{x}^{2}a{b}^{3}d{e}^{3}+108\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}-16\,x{a}^{3}b{e}^{4}-24\,x{a}^{2}{b}^{2}d{e}^{3}-48\,xa{b}^{3}{d}^{2}{e}^{2}+88\,x{b}^{4}{d}^{3}e-3\,{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}-6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-12\,a{b}^{3}{d}^{3}e+25\,{b}^{4}{d}^{4}}{12\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^5,x)

[Out]

1/12*((b*x+a)^2)^(3/2)*(12*ln(e*x+d)*x^4*b^4*e^4+48*ln(e*x+d)*x^3*b^4*d*e^3+72*l
n(e*x+d)*x^2*b^4*d^2*e^2-48*x^3*a*b^3*e^4+48*x^3*b^4*d*e^3+48*ln(e*x+d)*x*b^4*d^
3*e-36*x^2*a^2*b^2*e^4-72*x^2*a*b^3*d*e^3+108*x^2*b^4*d^2*e^2+12*ln(e*x+d)*b^4*d
^4-16*x*a^3*b*e^4-24*x*a^2*b^2*d*e^3-48*x*a*b^3*d^2*e^2+88*x*b^4*d^3*e-3*a^4*e^4
-4*a^3*b*d*e^3-6*a^2*b^2*d^2*e^2-12*a*b^3*d^3*e+25*b^4*d^4)/(b*x+a)^3/e^5/(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)^(3/2)*(b*x + a)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286681, size = 362, normalized size = 1.47 \[ \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, b^{4} d e^{3} x^{3} + 6 \, b^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} d^{3} e x + b^{4} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/12*(25*b^4*d^4 - 12*a*b^3*d^3*e - 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 - 3*a^4*e^
4 + 48*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 36*(3*b^4*d^2*e^2 - 2*a*b^3*d*e^3 - a^2*b^2
*e^4)*x^2 + 8*(11*b^4*d^3*e - 6*a*b^3*d^2*e^2 - 3*a^2*b^2*d*e^3 - 2*a^3*b*e^4)*x
+ 12*(b^4*e^4*x^4 + 4*b^4*d*e^3*x^3 + 6*b^4*d^2*e^2*x^2 + 4*b^4*d^3*e*x + b^4*d
^4)*log(e*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283231, size = 362, normalized size = 1.47 \[ b^{4} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (48 \,{\left (b^{4} d e^{2}{\rm sign}\left (b x + a\right ) - a b^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e{\rm sign}\left (b x + a\right ) - 2 \, a b^{3} d e^{2}{\rm sign}\left (b x + a\right ) - a^{2} b^{2} e^{3}{\rm sign}\left (b x + a\right )\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3}{\rm sign}\left (b x + a\right ) - 6 \, a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 3 \, a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - 2 \, a^{3} b e^{3}{\rm sign}\left (b x + a\right )\right )} x +{\left (25 \, b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 12 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) - 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) - 3 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-4\right )}}{12 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^4*e^(-5)*ln(abs(x*e + d))*sign(b*x + a) + 1/12*(48*(b^4*d*e^2*sign(b*x + a) -
a*b^3*e^3*sign(b*x + a))*x^3 + 36*(3*b^4*d^2*e*sign(b*x + a) - 2*a*b^3*d*e^2*sig
n(b*x + a) - a^2*b^2*e^3*sign(b*x + a))*x^2 + 8*(11*b^4*d^3*sign(b*x + a) - 6*a*
b^3*d^2*e*sign(b*x + a) - 3*a^2*b^2*d*e^2*sign(b*x + a) - 2*a^3*b*e^3*sign(b*x +
a))*x + (25*b^4*d^4*sign(b*x + a) - 12*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) - 3*a^4*e^4*sign(b*x + a))*e^(
-1))*e^(-4)/(x*e + d)^4