3.1570 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=48 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0708958, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.97975, size = 44, normalized size = 0.92 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 \left (d + e x\right )^{6} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*(d + e*x)**6*(a*e - b*d))

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Mathematica [B]  time = 0.149723, size = 218, normalized size = 4.54 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^2 b^3 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+a b^4 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5*e^5 + a^4*b*e^4*(d + 6*e*x) + a^3*b^2*e^3*(d^2 + 6*d*e*
x + 15*e^2*x^2) + a^2*b^3*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + a*
b^4*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + b^5*(d^5
+ 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)))/(6*e
^6*(a + b*x)*(d + e*x)^6)

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Maple [B]  time = 0.011, size = 283, normalized size = 5.9 \[ -{\frac{6\,{x}^{5}{b}^{5}{e}^{5}+15\,{x}^{4}a{b}^{4}{e}^{5}+15\,{x}^{4}{b}^{5}d{e}^{4}+20\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20\,{x}^{3}a{b}^{4}d{e}^{4}+20\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+15\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+15\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+15\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+15\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+6\,x{a}^{4}b{e}^{5}+6\,x{a}^{3}{b}^{2}d{e}^{4}+6\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+6\,xa{b}^{4}{d}^{3}{e}^{2}+6\,x{b}^{5}{d}^{4}e+{a}^{5}{e}^{5}+{a}^{4}bd{e}^{4}+{a}^{3}{b}^{2}{d}^{2}{e}^{3}+{a}^{2}{b}^{3}{d}^{3}{e}^{2}+a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{6\, \left ( ex+d \right ) ^{6}{e}^{6} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x)

[Out]

-1/6*(6*b^5*e^5*x^5+15*a*b^4*e^5*x^4+15*b^5*d*e^4*x^4+20*a^2*b^3*e^5*x^3+20*a*b^
4*d*e^4*x^3+20*b^5*d^2*e^3*x^3+15*a^3*b^2*e^5*x^2+15*a^2*b^3*d*e^4*x^2+15*a*b^4*
d^2*e^3*x^2+15*b^5*d^3*e^2*x^2+6*a^4*b*e^5*x+6*a^3*b^2*d*e^4*x+6*a^2*b^3*d^2*e^3
*x+6*a*b^4*d^3*e^2*x+6*b^5*d^4*e*x+a^5*e^5+a^4*b*d*e^4+a^3*b^2*d^2*e^3+a^2*b^3*d
^3*e^2+a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^6/e^6/(b*x+a)^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)^(5/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210302, size = 405, normalized size = 8.44 \[ -\frac{6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \,{\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*e^5*x^5 + b^5*d^5 + a*b^4*d^4*e + a^2*b^3*d^3*e^2 + a^3*b^2*d^2*e^3
+ a^4*b*d*e^4 + a^5*e^5 + 15*(b^5*d*e^4 + a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 + a*b
^4*d*e^4 + a^2*b^3*e^5)*x^3 + 15*(b^5*d^3*e^2 + a*b^4*d^2*e^3 + a^2*b^3*d*e^4 +
a^3*b^2*e^5)*x^2 + 6*(b^5*d^4*e + a*b^4*d^3*e^2 + a^2*b^3*d^2*e^3 + a^3*b^2*d*e^
4 + a^4*b*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 +
15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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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**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222258, size = 508, normalized size = 10.58 \[ -\frac{{\left (6 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 15 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 20 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5*e^5*sign(b*x + a) + 15*b^5*d*x^4*e^4*sign(b*x + a) + 20*b^5*d^2*
x^3*e^3*sign(b*x + a) + 15*b^5*d^3*x^2*e^2*sign(b*x + a) + 6*b^5*d^4*x*e*sign(b*
x + a) + b^5*d^5*sign(b*x + a) + 15*a*b^4*x^4*e^5*sign(b*x + a) + 20*a*b^4*d*x^3
*e^4*sign(b*x + a) + 15*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 6*a*b^4*d^3*x*e^2*sign
(b*x + a) + a*b^4*d^4*e*sign(b*x + a) + 20*a^2*b^3*x^3*e^5*sign(b*x + a) + 15*a^
2*b^3*d*x^2*e^4*sign(b*x + a) + 6*a^2*b^3*d^2*x*e^3*sign(b*x + a) + a^2*b^3*d^3*
e^2*sign(b*x + a) + 15*a^3*b^2*x^2*e^5*sign(b*x + a) + 6*a^3*b^2*d*x*e^4*sign(b*
x + a) + a^3*b^2*d^2*e^3*sign(b*x + a) + 6*a^4*b*x*e^5*sign(b*x + a) + a^4*b*d*e
^4*sign(b*x + a) + a^5*e^5*sign(b*x + a))*e^(-6)/(x*e + d)^6