3.1981 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=41 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (d+e x)^5 (b d-a e)} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(5*(b*d - a*e)*(d + e*x)^5)

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Rubi [A]  time = 0.10383, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (d+e x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(5*(b*d - a*e)*(d + e*x)^5)

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Rubi in Sympy [A]  time = 12.9968, size = 36, normalized size = 0.88 \[ - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 \left (d + e x\right )^{5} \left (a e - b d\right )} \]

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)**6,x)

[Out]

-(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*(d + e*x)**5*(a*e - b*d))

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Mathematica [B]  time = 0.130341, size = 158, normalized size = 3.85 \[ -\frac{\sqrt{(a+b x)^2} \left (a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )}{5 e^5 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*
x + 10*e^2*x^2) + a*b^3*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d
^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)))/(5*e^5*(a + b*x)*(
d + e*x)^5)

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Maple [B]  time = 0.012, size = 197, normalized size = 4.8 \[ -{\frac{5\,{x}^{4}{b}^{4}{e}^{4}+10\,{x}^{3}a{b}^{3}{e}^{4}+10\,{x}^{3}{b}^{4}d{e}^{3}+10\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+10\,{x}^{2}a{b}^{3}d{e}^{3}+10\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+5\,x{a}^{3}b{e}^{4}+5\,x{a}^{2}{b}^{2}d{e}^{3}+5\,xa{b}^{3}{d}^{2}{e}^{2}+5\,x{b}^{4}{d}^{3}e+{a}^{4}{e}^{4}+{a}^{3}bd{e}^{3}+{a}^{2}{b}^{2}{d}^{2}{e}^{2}+a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{5\, \left ( ex+d \right ) ^{5}{e}^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^6,x)

[Out]

-1/5*(5*b^4*e^4*x^4+10*a*b^3*e^4*x^3+10*b^4*d*e^3*x^3+10*a^2*b^2*e^4*x^2+10*a*b^
3*d*e^3*x^2+10*b^4*d^2*e^2*x^2+5*a^3*b*e^4*x+5*a^2*b^2*d*e^3*x+5*a*b^3*d^2*e^2*x
+5*b^4*d^3*e*x+a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^4)*((b*x+a)
^2)^(3/2)/(e*x+d)^5/e^5/(b*x+a)^3

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281519, size = 290, normalized size = 7.07 \[ -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^
4*e^4 + 10*(b^4*d*e^3 + a*b^3*e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2
*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a^3*b*e^4)*x)/(e^10*x
^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28385, size = 351, normalized size = 8.56 \[ -\frac{{\left (5 \, b^{4} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 10 \, b^{4} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 10 \, b^{4} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, b^{4} d^{3} x e{\rm sign}\left (b x + a\right ) + b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 10 \, a b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 10 \, a b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 5 \, a^{2} b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a^{3} b x e^{4}{\rm sign}\left (b x + a\right ) + a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{5 \,{\left (x e + d\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

-1/5*(5*b^4*x^4*e^4*sign(b*x + a) + 10*b^4*d*x^3*e^3*sign(b*x + a) + 10*b^4*d^2*
x^2*e^2*sign(b*x + a) + 5*b^4*d^3*x*e*sign(b*x + a) + b^4*d^4*sign(b*x + a) + 10
*a*b^3*x^3*e^4*sign(b*x + a) + 10*a*b^3*d*x^2*e^3*sign(b*x + a) + 5*a*b^3*d^2*x*
e^2*sign(b*x + a) + a*b^3*d^3*e*sign(b*x + a) + 10*a^2*b^2*x^2*e^4*sign(b*x + a)
+ 5*a^2*b^2*d*x*e^3*sign(b*x + a) + a^2*b^2*d^2*e^2*sign(b*x + a) + 5*a^3*b*x*e
^4*sign(b*x + a) + a^3*b*d*e^3*sign(b*x + a) + a^4*e^4*sign(b*x + a))*e^(-5)/(x*
e + d)^5