3.1555 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=143 \[ \frac{b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{60 (d+e x)^4 (b d-a e)^3}+\frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{15 (d+e x)^5 (b d-a e)^2}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{6 (d+e x)^6 (b d-a e)} \]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(6*(b*d - a*e)*(d + e*x)^6) + (b*(a
+ b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(15*(b*d - a*e)^2*(d + e*x)^5) + (b^2*(a
+ b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(60*(b*d - a*e)^3*(d + e*x)^4)

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Rubi [A]  time = 0.259333, antiderivative size = 200, normalized size of antiderivative = 1.4, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^4 (a+b x) (d+e x)^4}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^4 (a+b x) (d+e x)^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{6 e^4 (a+b x) (d+e x)^6}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.0723, size = 131, normalized size = 0.92 \[ - \frac{b e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{60 \left (d + e x\right )^{5} \left (a e - b d\right )^{3}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{24 \left (d + e x\right )^{5} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 \left (d + e x\right )^{6} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**7,x)

[Out]

-b*e*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(60*(d + e*x)**5*(a*e - b*d)**3) + b*(2
*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(24*(d + e*x)**5*(a*e - b*d)**2)
- (2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*(d + e*x)**6*(a*e - b*d
))

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Mathematica [A]  time = 0.0825764, size = 112, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d
*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)))/(60*e^4
*(a + b*x)*(d + e*x)^6)

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Maple [A]  time = 0.011, size = 131, normalized size = 0.9 \[ -{\frac{20\,{x}^{3}{b}^{3}{e}^{3}+45\,{x}^{2}a{b}^{2}{e}^{3}+15\,{x}^{2}{b}^{3}d{e}^{2}+36\,x{a}^{2}b{e}^{3}+18\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e+10\,{a}^{3}{e}^{3}+6\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{60\,{e}^{4} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x)

[Out]

-1/60/e^4*(20*b^3*e^3*x^3+45*a*b^2*e^3*x^2+15*b^3*d*e^2*x^2+36*a^2*b*e^3*x+18*a*
b^2*d*e^2*x+6*b^3*d^2*e*x+10*a^3*e^3+6*a^2*b*d*e^2+3*a*b^2*d^2*e+b^3*d^3)*((b*x+
a)^2)^(3/2)/(e*x+d)^6/(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)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208655, size = 231, normalized size = 1.62 \[ -\frac{20 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 10 \, a^{3} e^{3} + 15 \,{\left (b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + 6 \, a^{2} b e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*e^3*x^3 + b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2 + 10*a^3*e^3 + 1
5*(b^3*d*e^2 + 3*a*b^2*e^3)*x^2 + 6*(b^3*d^2*e + 3*a*b^2*d*e^2 + 6*a^2*b*e^3)*x)
/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*
d^5*e^5*x + d^6*e^4)

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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)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21593, size = 228, normalized size = 1.59 \[ -\frac{{\left (20 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 45 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 18 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 36 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*x^3*e^3*sign(b*x + a) + 15*b^3*d*x^2*e^2*sign(b*x + a) + 6*b^3*d^2
*x*e*sign(b*x + a) + b^3*d^3*sign(b*x + a) + 45*a*b^2*x^2*e^3*sign(b*x + a) + 18
*a*b^2*d*x*e^2*sign(b*x + a) + 3*a*b^2*d^2*e*sign(b*x + a) + 36*a^2*b*x*e^3*sign
(b*x + a) + 6*a^2*b*d*e^2*sign(b*x + a) + 10*a^3*e^3*sign(b*x + a))*e^(-4)/(x*e
+ d)^6