3.2098 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^3 (a+b x)}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^3 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3 (a+b x) (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.20876, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^3 (a+b x)}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^3 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4826, size = 124, normalized size = 0.84 \[ \frac{8 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2} \sqrt{d + e x}} - \frac{16 b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{3} \left (a + b x\right ) \sqrt{d + e x}} - \frac{2 \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**(5/2),x)

[Out]

8*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**2*sqrt(d + e*x)) - 16*b*(a*e - b*d)*s
qrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**3*(a + b*x)*sqrt(d + e*x)) - 2*(a + b*x)*s
qrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.082354, size = 79, normalized size = 0.53 \[ -\frac{2 \sqrt{(a+b x)^2} \left (a^2 e^2+2 a b e (2 d+3 e x)+b^2 \left (-\left (8 d^2+12 d e x+3 e^2 x^2\right )\right )\right )}{3 e^3 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(a^2*e^2 + 2*a*b*e*(2*d + 3*e*x) - b^2*(8*d^2 + 12*d*e*x +
3*e^2*x^2)))/(3*e^3*(a + b*x)*(d + e*x)^(3/2))

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Maple [A]  time = 0.009, size = 78, normalized size = 0.5 \[ -{\frac{-6\,{x}^{2}{b}^{2}{e}^{2}+12\,xab{e}^{2}-24\,x{b}^{2}de+2\,{a}^{2}{e}^{2}+8\,abde-16\,{b}^{2}{d}^{2}}{3\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.72796, size = 130, normalized size = 0.88 \[ -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )} a}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (3 \, b e^{2} x^{2} + 8 \, b d^{2} - 2 \, a d e + 3 \,{\left (4 \, b d e - a e^{2}\right )} x\right )} b}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*b*e*x + 2*b*d + a*e)*a/((e^3*x + d*e^2)*sqrt(e*x + d)) + 2/3*(3*b*e^2*x^
2 + 8*b*d^2 - 2*a*d*e + 3*(4*b*d*e - a*e^2)*x)*b/((e^4*x + d*e^3)*sqrt(e*x + d))

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Fricas [A]  time = 0.278479, size = 100, normalized size = 0.68 \[ \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 4 \, a b d e - a^{2} e^{2} + 6 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.32944, size = 150, normalized size = 1.01 \[ 2 \, \sqrt{x e + d} b^{2} e^{\left (-3\right )}{\rm sign}\left (b x + a\right ) + \frac{2 \,{\left (6 \,{\left (x e + d\right )} b^{2} d{\rm sign}\left (b x + a\right ) - b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 6 \,{\left (x e + d\right )} a b e{\rm sign}\left (b x + a\right ) + 2 \, a b d e{\rm sign}\left (b x + a\right ) - a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*b^2*e^(-3)*sign(b*x + a) + 2/3*(6*(x*e + d)*b^2*d*sign(b*x + a)
- b^2*d^2*sign(b*x + a) - 6*(x*e + d)*a*b*e*sign(b*x + a) + 2*a*b*d*e*sign(b*x +
a) - a^2*e^2*sign(b*x + a))*e^(-3)/(x*e + d)^(3/2)