3.1977 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=239 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) (d+e x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^5 (a+b x)}+\frac{6 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^5 (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.401919, antiderivative size = 239, 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 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) (d+e x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^5 (a+b x)}+\frac{6 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.6427, size = 187, normalized size = 0.78 \[ \frac{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{2 b \left (3 a + 3 b x\right ) \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{3}} + \frac{4 b \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{4}} + \frac{4 b \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )} \]

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)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.272615, size = 183, normalized size = 0.77 \[ \frac{\sqrt{(a+b x)^2} \left (-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )-12 b (d+e x) (b d-a e)^3 \log (d+e x)+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )}{3 e^5 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x +
e^2*x^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 +
9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) - 12*b*(b*d - a*e)^3*(d + e*
x)*Log[d + e*x]))/(3*e^5*(a + b*x)*(d + e*x))

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 327, normalized size = 1.4 \[{\frac{{x}^{4}{b}^{4}{e}^{4}+6\,{x}^{3}a{b}^{3}{e}^{4}-2\,{x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( ex+d \right ) x{a}^{3}b{e}^{4}-36\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}+36\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}-12\,\ln \left ( ex+d \right ) x{b}^{4}{d}^{3}e+18\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-18\,{x}^{2}a{b}^{3}d{e}^{3}+6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{3}bd{e}^{3}-36\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+36\,\ln \left ( ex+d \right ) a{b}^{3}{d}^{3}e-12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}+18\,x{a}^{2}{b}^{2}d{e}^{3}-24\,xa{b}^{3}{d}^{2}{e}^{2}+9\,x{b}^{4}{d}^{3}e-3\,{a}^{4}{e}^{4}+12\,{a}^{3}bd{e}^{3}-18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,a{b}^{3}{d}^{3}e-3\,{b}^{4}{d}^{4}}{3\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^2,x)

[Out]

1/3*((b*x+a)^2)^(3/2)*(x^4*b^4*e^4+6*x^3*a*b^3*e^4-2*x^3*b^4*d*e^3+12*ln(e*x+d)*
x*a^3*b*e^4-36*ln(e*x+d)*x*a^2*b^2*d*e^3+36*ln(e*x+d)*x*a*b^3*d^2*e^2-12*ln(e*x+
d)*x*b^4*d^3*e+18*x^2*a^2*b^2*e^4-18*x^2*a*b^3*d*e^3+6*x^2*b^4*d^2*e^2+12*ln(e*x
+d)*a^3*b*d*e^3-36*ln(e*x+d)*a^2*b^2*d^2*e^2+36*ln(e*x+d)*a*b^3*d^3*e-12*ln(e*x+
d)*b^4*d^4+18*x*a^2*b^2*d*e^3-24*x*a*b^3*d^2*e^2+9*x*b^4*d^3*e-3*a^4*e^4+12*a^3*
b*d*e^3-18*a^2*b^2*d^2*e^2+12*a*b^3*d^3*e-3*b^4*d^4)/(b*x+a)^3/e^5/(e*x+d)

_______________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.288507, size = 360, normalized size = 1.51 \[ \frac{b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 2 \,{\left (b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (3 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3}\right )} x - 12 \,{\left (b^{4} d^{4} - 3 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} - a^{3} b d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.293159, size = 362, normalized size = 1.51 \[ -4 \,{\left (b^{4} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} b e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{4} x^{3} e^{4}{\rm sign}\left (b x + a\right ) - 3 \, b^{4} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 9 \, b^{4} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a b^{3} x^{2} e^{4}{\rm sign}\left (b x + a\right ) - 24 \, a b^{3} d x e^{3}{\rm sign}\left (b x + a\right ) + 18 \, a^{2} b^{2} x e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )} - \frac{{\left (b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{x e + d} \]

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)^2,x, algorithm="giac")

[Out]

-4*(b^4*d^3*sign(b*x + a) - 3*a*b^3*d^2*e*sign(b*x + a) + 3*a^2*b^2*d*e^2*sign(b
*x + a) - a^3*b*e^3*sign(b*x + a))*e^(-5)*ln(abs(x*e + d)) + 1/3*(b^4*x^3*e^4*si
gn(b*x + a) - 3*b^4*d*x^2*e^3*sign(b*x + a) + 9*b^4*d^2*x*e^2*sign(b*x + a) + 6*
a*b^3*x^2*e^4*sign(b*x + a) - 24*a*b^3*d*x*e^3*sign(b*x + a) + 18*a^2*b^2*x*e^4*
sign(b*x + a))*e^(-6) - (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)/(x*e + d)