3.1665 \(\int (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^2 (a+b x)} \]

[Out]

(-2*(b*d - a*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^2*(a + b*x))
+ (2*b*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^2*(a + b*x))

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Rubi [A]  time = 0.123292, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^2*(a + b*x))
+ (2*b*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^2*(a + b*x))

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Rubi in Sympy [A]  time = 11.9904, size = 80, normalized size = 0.83 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9 e} + \frac{4 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(7/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9*e) + 4*(d + e*x)**(7/2)*(
a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**2*(a + b*x))

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Mathematica [A]  time = 0.081307, size = 48, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} (9 a e-2 b d+7 b e x)}{63 e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(-2*b*d + 9*a*e + 7*b*e*x))/(63*e^2*(a + b*
x))

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Maple [A]  time = 0.004, size = 43, normalized size = 0.5 \[{\frac{14\,bex+18\,ae-4\,bd}{63\,{e}^{2} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(e*x+d)^(7/2)*(7*b*e*x+9*a*e-2*b*d)*((b*x+a)^2)^(1/2)/e^2/(b*x+a)

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Maxima [A]  time = 0.764251, size = 126, normalized size = 1.31 \[ \frac{2 \,{\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e +{\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \,{\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} +{\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{63 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*b*e^4*x^4 - 2*b*d^4 + 9*a*d^3*e + (19*b*d*e^3 + 9*a*e^4)*x^3 + 3*(5*b*d^
2*e^2 + 9*a*d*e^3)*x^2 + (b*d^3*e + 27*a*d^2*e^2)*x)*sqrt(e*x + d)/e^2

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Fricas [A]  time = 0.205407, size = 126, normalized size = 1.31 \[ \frac{2 \,{\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e +{\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \,{\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} +{\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{63 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*b*e^4*x^4 - 2*b*d^4 + 9*a*d^3*e + (19*b*d*e^3 + 9*a*e^4)*x^3 + 3*(5*b*d^
2*e^2 + 9*a*d*e^3)*x^2 + (b*d^3*e + 27*a*d^2*e^2)*x)*sqrt(e*x + d)/e^2

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221723, size = 356, normalized size = 3.71 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b d^{2} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2}{\rm sign}\left (b x + a\right ) + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b d e^{\left (-13\right )}{\rm sign}\left (b x + a\right ) + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d{\rm sign}\left (b x + a\right ) + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a e^{\left (-12\right )}{\rm sign}\left (b x + a\right ) +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b e^{\left (-25\right )}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*d^2*e^(-1)*sign(b*x + a) +
105*(x*e + d)^(3/2)*a*d^2*sign(b*x + a) + 6*(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)*b*d*e^(-13)*sign(b*x + a) + 42*
(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*d*sign(b*x + a) + 3*(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)*a*e^(-12)*si
gn(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)*b*e^(-25)*sign(b*x + a))*e^(-
1)