3.740 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=193 \[ \frac{6 a b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (6*a^2*
b*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^
(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(9/2)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^7*(a + b*x^2))

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Rubi [A]  time = 0.159123, antiderivative size = 193, 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 \[ \frac{6 a b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(5/2),x]

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (6*a^2*
b*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^
(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(9/2)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^7*(a + b*x^2))

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Rubi in Sympy [A]  time = 18.1344, size = 156, normalized size = 0.81 \[ - \frac{256 a^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{45 d \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )} + \frac{64 a^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{15 d \left (d x\right )^{\frac{3}{2}}} + \frac{8 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{15 d \left (d x\right )^{\frac{3}{2}}} + \frac{2 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{9 d \left (d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(5/2),x)

[Out]

-256*a**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(45*d*(d*x)**(3/2)*(a + b*x**2)) +
64*a**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(15*d*(d*x)**(3/2)) + 8*a*(a + b*x*
*2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(15*d*(d*x)**(3/2)) + 2*(a**2 + 2*a*b*x*
*2 + b**2*x**4)**(3/2)/(9*d*(d*x)**(3/2))

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Mathematica [A]  time = 0.0422326, size = 66, normalized size = 0.34 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (-15 a^3+135 a^2 b x^2+27 a b^2 x^4+5 b^3 x^6\right )}{45 (d x)^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(5/2),x]

[Out]

(2*x*Sqrt[(a + b*x^2)^2]*(-15*a^3 + 135*a^2*b*x^2 + 27*a*b^2*x^4 + 5*b^3*x^6))/(
45*(d*x)^(5/2)*(a + b*x^2))

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Maple [A]  time = 0.009, size = 61, normalized size = 0.3 \[ -{\frac{2\, \left ( -5\,{b}^{3}{x}^{6}-27\,a{x}^{4}{b}^{2}-135\,{a}^{2}b{x}^{2}+15\,{a}^{3} \right ) x}{45\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( dx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/(d*x)^(5/2),x)

[Out]

-2/45*x*(-5*b^3*x^6-27*a*b^2*x^4-135*a^2*b*x^2+15*a^3)*((b*x^2+a)^2)^(3/2)/(b*x^
2+a)^3/(d*x)^(5/2)

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Maxima [A]  time = 0.712538, size = 116, normalized size = 0.6 \[ \frac{2 \,{\left ({\left (5 \, b^{3} \sqrt{d} x^{3} + 9 \, a b^{2} \sqrt{d} x\right )} x^{\frac{3}{2}} + \frac{18 \,{\left (a b^{2} \sqrt{d} x^{3} + 5 \, a^{2} b \sqrt{d} x\right )}}{\sqrt{x}} + \frac{15 \,{\left (3 \, a^{2} b \sqrt{d} x^{3} - a^{3} \sqrt{d} x\right )}}{x^{\frac{5}{2}}}\right )}}{45 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

2/45*((5*b^3*sqrt(d)*x^3 + 9*a*b^2*sqrt(d)*x)*x^(3/2) + 18*(a*b^2*sqrt(d)*x^3 +
5*a^2*b*sqrt(d)*x)/sqrt(x) + 15*(3*a^2*b*sqrt(d)*x^3 - a^3*sqrt(d)*x)/x^(5/2))/d
^3

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Fricas [A]  time = 0.271187, size = 61, normalized size = 0.32 \[ \frac{2 \,{\left (5 \, b^{3} x^{6} + 27 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} - 15 \, a^{3}\right )}}{45 \, \sqrt{d x} d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/45*(5*b^3*x^6 + 27*a*b^2*x^4 + 135*a^2*b*x^2 - 15*a^3)/(sqrt(d*x)*d^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{\left (d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(5/2),x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/(d*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.26696, size = 142, normalized size = 0.74 \[ -\frac{2 \,{\left (\frac{15 \, a^{3} d{\rm sign}\left (b x^{2} + a\right )}{\sqrt{d x} x} - \frac{5 \, \sqrt{d x} b^{3} d^{36} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 27 \, \sqrt{d x} a b^{2} d^{36} x^{2}{\rm sign}\left (b x^{2} + a\right ) + 135 \, \sqrt{d x} a^{2} b d^{36}{\rm sign}\left (b x^{2} + a\right )}{d^{36}}\right )}}{45 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/45*(15*a^3*d*sign(b*x^2 + a)/(sqrt(d*x)*x) - (5*sqrt(d*x)*b^3*d^36*x^4*sign(b
*x^2 + a) + 27*sqrt(d*x)*a*b^2*d^36*x^2*sign(b*x^2 + a) + 135*sqrt(d*x)*a^2*b*d^
36*sign(b*x^2 + a))/d^36)/d^3