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

Optimal. Leaf size=191 \[ \frac{6 a b^2 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left (a+b x^2\right )}+\frac{2 a^2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )} \]

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*Sqrt[d*x]*(a + b*x^2)) + (2*a^2*b*(d
*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(7
/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(11/2)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^7*(a + b*x^2))

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Rubi [A]  time = 0.159989, antiderivative size = 191, 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 \[ \frac{6 a b^2 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^5 \left (a+b x^2\right )}+\frac{2 a^2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^7 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*Sqrt[d*x]*(a + b*x^2)) + (2*a^2*b*(d
*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(7
/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(11/2)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^7*(a + b*x^2))

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Rubi in Sympy [A]  time = 18.2303, size = 156, normalized size = 0.82 \[ - \frac{256 a^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{77 d \sqrt{d x} \left (a + b x^{2}\right )} + \frac{64 a^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{77 d \sqrt{d x}} + \frac{24 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{77 d \sqrt{d x}} + \frac{2 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{11 d \sqrt{d x}} \]

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)**(3/2),x)

[Out]

-256*a**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(77*d*sqrt(d*x)*(a + b*x**2)) + 64
*a**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(77*d*sqrt(d*x)) + 24*a*(a + b*x**2)*s
qrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(77*d*sqrt(d*x)) + 2*(a**2 + 2*a*b*x**2 + b**
2*x**4)**(3/2)/(11*d*sqrt(d*x))

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Mathematica [A]  time = 0.0414621, size = 66, normalized size = 0.35 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (-77 a^3+77 a^2 b x^2+33 a b^2 x^4+7 b^3 x^6\right )}{77 (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[(a + b*x^2)^2]*(-77*a^3 + 77*a^2*b*x^2 + 33*a*b^2*x^4 + 7*b^3*x^6))/(7
7*(d*x)^(3/2)*(a + b*x^2))

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Maple [A]  time = 0.009, size = 61, normalized size = 0.3 \[ -{\frac{2\, \left ( -7\,{b}^{3}{x}^{6}-33\,a{x}^{4}{b}^{2}-77\,{a}^{2}b{x}^{2}+77\,{a}^{3} \right ) x}{77\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( dx \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-2/77*x*(-7*b^3*x^6-33*a*b^2*x^4-77*a^2*b*x^2+77*a^3)*((b*x^2+a)^2)^(3/2)/(b*x^2
+a)^3/(d*x)^(3/2)

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Maxima [A]  time = 0.721661, size = 117, normalized size = 0.61 \[ \frac{2 \,{\left (3 \,{\left (7 \, b^{3} \sqrt{d} x^{3} + 11 \, a b^{2} \sqrt{d} x\right )} x^{\frac{5}{2}} + 22 \,{\left (3 \, a b^{2} \sqrt{d} x^{3} + 7 \, a^{2} b \sqrt{d} x\right )} \sqrt{x} + \frac{77 \,{\left (a^{2} b \sqrt{d} x^{3} - 3 \, a^{3} \sqrt{d} x\right )}}{x^{\frac{3}{2}}}\right )}}{231 \, d^{2}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/231*(3*(7*b^3*sqrt(d)*x^3 + 11*a*b^2*sqrt(d)*x)*x^(5/2) + 22*(3*a*b^2*sqrt(d)*
x^3 + 7*a^2*b*sqrt(d)*x)*sqrt(x) + 77*(a^2*b*sqrt(d)*x^3 - 3*a^3*sqrt(d)*x)/x^(3
/2))/d^2

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Fricas [A]  time = 0.274854, size = 57, normalized size = 0.3 \[ \frac{2 \,{\left (7 \, b^{3} x^{6} + 33 \, a b^{2} x^{4} + 77 \, a^{2} b x^{2} - 77 \, a^{3}\right )}}{77 \, \sqrt{d x} d} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/77*(7*b^3*x^6 + 33*a*b^2*x^4 + 77*a^2*b*x^2 - 77*a^3)/(sqrt(d*x)*d)

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

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267531, size = 138, normalized size = 0.72 \[ -\frac{2 \,{\left (\frac{77 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{\sqrt{d x}} - \frac{7 \, \sqrt{d x} b^{3} d^{65} x^{5}{\rm sign}\left (b x^{2} + a\right ) + 33 \, \sqrt{d x} a b^{2} d^{65} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 77 \, \sqrt{d x} a^{2} b d^{65} x{\rm sign}\left (b x^{2} + a\right )}{d^{66}}\right )}}{77 \, d} \]

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)^(3/2),x, algorithm="giac")

[Out]

-2/77*(77*a^3*sign(b*x^2 + a)/sqrt(d*x) - (7*sqrt(d*x)*b^3*d^65*x^5*sign(b*x^2 +
a) + 33*sqrt(d*x)*a*b^2*d^65*x^3*sign(b*x^2 + a) + 77*sqrt(d*x)*a^2*b*d^65*x*si
gn(b*x^2 + a))/d^66)/d