3.2129 \(\int \frac{a+b x}{(d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.119669, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{2 (a+b x)}{5 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2537, size = 41, normalized size = 1. \[ - \frac{2 \left (a + b x\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0367209, size = 32, normalized size = 0.78 \[ -\frac{2 (a+b x)}{5 e \sqrt{(a+b x)^2} (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{5\,e} \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x)

[Out]

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

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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*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282898, size = 42, normalized size = 1.02 \[ -\frac{2}{5 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5/((e^3*x^2 + 2*d*e^2*x + d^2*e)*sqrt(e*x + d))

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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)/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282207, size = 24, normalized size = 0.59 \[ -\frac{2 \, e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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