3.2026 \(\int \frac{(a+b x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=162 \[ -\frac{(d+e x)^3}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) (d+e x)^2}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) (b d-a e)^2 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e^2 x (a+b x) (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.272829, antiderivative size = 162, 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 \[ -\frac{(d+e x)^3}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) (d+e x)^2}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) (b d-a e)^2 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e^2 x (a+b x) (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.5422, size = 155, normalized size = 0.96 \[ - \frac{\left (d + e x\right )^{3}}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{3 e \left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{3 e^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{4}} + \frac{3 e \left (a + b x\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.119229, size = 120, normalized size = 0.74 \[ \frac{2 a^3 e^3-2 a^2 b e^2 (3 d+2 e x)+3 a b^2 e \left (2 d^2+2 d e x-e^2 x^2\right )+6 e (a+b x) (b d-a e)^2 \log (a+b x)+b^3 \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )}{2 b^4 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 209, normalized size = 1.3 \[{\frac{ \left ({x}^{3}{b}^{3}{e}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}+6\,\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}e-3\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ){a}^{3}{e}^{3}-12\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}+6\,\ln \left ( bx+a \right ) a{b}^{2}{d}^{2}e-4\,x{a}^{2}b{e}^{3}+6\,xa{b}^{2}d{e}^{2}+2\,{a}^{3}{e}^{3}-6\,{a}^{2}bd{e}^{2}+6\,a{b}^{2}{d}^{2}e-2\,{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) ^{2}}{2\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.706968, size = 780, normalized size = 4.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e^3*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b) - 5/2*a*e^3*x^2/(sqrt(b^2*x^2 + 2*
a*b*x + a^2)*b^2) + 6*a^2*e^3*log(x + a/b)/((b^2)^(3/2)*b) + 9*a^4*b*e^3/((b^2)^
(7/2)*(x + a/b)^2) + 12*a^3*e^3*x/((b^2)^(5/2)*(x + a/b)^2) - 5*a^3*e^3/(sqrt(b^
2*x^2 + 2*a*b*x + a^2)*b^4) - 1/2*a*d^3/((b^2)^(3/2)*(x + a/b)^2) + 5/2*a^4*e^3/
((b^2)^(3/2)*b^3*(x + a/b)^2) + (3*b*d*e^2 + a*e^3)*x^2/(sqrt(b^2*x^2 + 2*a*b*x
+ a^2)*b^2) + 3*(b*d^2*e + a*d*e^2)*log(x + a/b)/(b^2)^(3/2) - 3*(3*b*d*e^2 + a*
e^3)*a*log(x + a/b)/((b^2)^(3/2)*b) - 9/2*(3*b*d*e^2 + a*e^3)*a^3*b/((b^2)^(7/2)
*(x + a/b)^2) + 9/2*(b*d^2*e + a*d*e^2)*a^2*b^2/((b^2)^(7/2)*(x + a/b)^2) - 6*(3
*b*d*e^2 + a*e^3)*a^2*x/((b^2)^(5/2)*(x + a/b)^2) + 6*(b*d^2*e + a*d*e^2)*a*b*x/
((b^2)^(5/2)*(x + a/b)^2) + 2*(3*b*d*e^2 + a*e^3)*a^2/(sqrt(b^2*x^2 + 2*a*b*x +
a^2)*b^4) - (b*d^3 + 3*a*d^2*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - (3*b*d*e^2
+ a*e^3)*a^3/((b^2)^(3/2)*b^3*(x + a/b)^2) + 1/2*(b*d^3 + 3*a*d^2*e)*a/((b^2)^(
3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 0.279902, size = 234, normalized size = 1.44 \[ \frac{b^{3} e^{3} x^{3} - 2 \, b^{3} d^{3} + 6 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} + 3 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^3*e^3*x^3 - 2*b^3*d^3 + 6*a*b^2*d^2*e - 6*a^2*b*d*e^2 + 2*a^3*e^3 + 3*(2*
b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(3*a*b^2*d*e^2 - 2*a^2*b*e^3)*x + 6*(a*b^2*d^2*e
- 2*a^2*b*d*e^2 + a^3*e^3 + (b^3*d^2*e - 2*a*b^2*d*e^2 + a^2*b*e^3)*x)*log(b*x +
a))/(b^5*x + a*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{3}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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