3.2122 \(\int \frac{(a+b x) (d+e x)^{7/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(a + b*x)*(d + e*x)^(9/2))/(9*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.121748, 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 \[ \frac{2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

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)*(d + e*x)^(9/2))/(9*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 16.2547, size = 37, normalized size = 0.9 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{9}{2}}}{9 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

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)*(d + e*x)**(9/2)/(9*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0586225, size = 32, normalized size = 0.78 \[ \frac{2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt{(a+b x)^2}} \]

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)*(d + e*x)^(9/2))/(9*e*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.004, size = 27, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{9\,e} \left ( ex+d \right ) ^{{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

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]

2/9*(b*x+a)*(e*x+d)^(9/2)/e/((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)*(e*x + d)^(7/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28143, size = 68, normalized size = 1.66 \[ \frac{2 \,{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \sqrt{e x + d}}{9 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)*sqrt(e*x + d)/e

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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 = 0.287421, size = 235, normalized size = 5.73 \[ \frac{2}{315} \,{\left (105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}{\rm sign}\left (b x + a\right ) + 63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} d^{2}{\rm sign}\left (b x + a\right ) + 9 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} d e^{\left (-12\right )}{\rm sign}\left (b x + a\right ) +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} e^{\left (-24\right )}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(105*(x*e + d)^(3/2)*d^3*sign(b*x + a) + 63*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*d^2*sign(b*x + a) + 9*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*
d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*d*e^(-12)*sign(b*x + a) + (35*(x*e + d)^(9
/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e
+ d)^(3/2)*d^3*e^24)*e^(-24)*sign(b*x + a))*e^(-1)