3.2099 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.212478, antiderivative size = 150, 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 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.3921, size = 124, normalized size = 0.83 \[ - \frac{8 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 e^{3} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

-8*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e**2*(d + e*x)**(3/2)) + 16*b*(a*e - b*
d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*e**3*(a + b*x)*(d + e*x)**(3/2)) - 2*(a
+ b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.0839274, size = 79, normalized size = 0.53 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(3*a^2*e^2 + 2*a*b*e*(2*d + 5*e*x) + b^2*(8*d^2 + 20*d*e*x
+ 15*e^2*x^2)))/(15*e^3*(a + b*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.01, size = 79, normalized size = 0.5 \[ -{\frac{30\,{x}^{2}{b}^{2}{e}^{2}+20\,xab{e}^{2}+40\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+8\,abde+16\,{b}^{2}{d}^{2}}{15\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/(e*x+d)^(5/2)*(15*b^2*e^2*x^2+10*a*b*e^2*x+20*b^2*d*e*x+3*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 = 0.720543, size = 159, normalized size = 1.06 \[ -\frac{2 \,{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} a}{15 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )} \sqrt{e x + d}} - \frac{2 \,{\left (15 \, b e^{2} x^{2} + 8 \, b d^{2} + 2 \, a d e + 5 \,{\left (4 \, b d e + a e^{2}\right )} x\right )} b}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(5*b*e*x + 2*b*d + 3*a*e)*a/((e^4*x^2 + 2*d*e^3*x + d^2*e^2)*sqrt(e*x + d)
) - 2/15*(15*b*e^2*x^2 + 8*b*d^2 + 2*a*d*e + 5*(4*b*d*e + a*e^2)*x)*b/((e^5*x^2
+ 2*d*e^4*x + d^2*e^3)*sqrt(e*x + d))

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Fricas [A]  time = 0.287128, size = 113, normalized size = 0.75 \[ -\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 4 \, a b d e + 3 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*b^2*e^2*x^2 + 8*b^2*d^2 + 4*a*b*d*e + 3*a^2*e^2 + 10*(2*b^2*d*e + a*b*
e^2)*x)/((e^5*x^2 + 2*d*e^4*x + d^2*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308948, size = 146, normalized size = 0.97 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{2}{\rm sign}\left (b x + a\right ) - 10 \,{\left (x e + d\right )} b^{2} d{\rm sign}\left (b x + a\right ) + 3 \, b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 10 \,{\left (x e + d\right )} a b e{\rm sign}\left (b x + a\right ) - 6 \, a b d e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*(x*e + d)^2*b^2*sign(b*x + a) - 10*(x*e + d)*b^2*d*sign(b*x + a) + 3*b
^2*d^2*sign(b*x + a) + 10*(x*e + d)*a*b*e*sign(b*x + a) - 6*a*b*d*e*sign(b*x + a
) + 3*a^2*e^2*sign(b*x + a))*e^(-3)/(x*e + d)^(5/2)