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

Optimal. Leaf size=195 \[ \frac{6 a b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

[Out]

(2*a^3*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (6*a^2*b
*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^3*(a + b*x^2)) + (6*a*b^2*(d*
x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(
15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*d^7*(a + b*x^2))

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Rubi [A]  time = 0.156643, antiderivative size = 195, 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 \[ \frac{6 a b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (6*a^2*b
*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d^3*(a + b*x^2)) + (6*a*b^2*(d*
x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(
15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*d^7*(a + b*x^2))

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Rubi in Sympy [A]  time = 18.1618, size = 156, normalized size = 0.8 \[ \frac{256 a^{3} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1155 d \left (a + b x^{2}\right )} + \frac{64 a^{2} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{385 d} + \frac{8 a \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{55 d} + \frac{2 \left (d x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{15 d} \]

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)**(1/2),x)

[Out]

256*a**3*(d*x)**(3/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(1155*d*(a + b*x**2))
+ 64*a**2*(d*x)**(3/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(385*d) + 8*a*(d*x)**
(3/2)*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(55*d) + 2*(d*x)**(3/2)*(
a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(15*d)

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Mathematica [A]  time = 0.0352154, size = 66, normalized size = 0.34 \[ \frac{2 \sqrt{d x} \sqrt{\left (a+b x^2\right )^2} \left (385 a^3 x+495 a^2 b x^3+315 a b^2 x^5+77 b^3 x^7\right )}{1155 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(385*a^3*x + 495*a^2*b*x^3 + 315*a*b^2*x^5 + 77
*b^3*x^7))/(1155*(a + b*x^2))

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Maple [A]  time = 0.008, size = 61, normalized size = 0.3 \[{\frac{2\,x \left ( 77\,{b}^{3}{x}^{6}+315\,a{x}^{4}{b}^{2}+495\,{a}^{2}b{x}^{2}+385\,{a}^{3} \right ) }{1155\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}\sqrt{dx}} \]

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)^(1/2),x)

[Out]

2/1155*x*(77*b^3*x^6+315*a*b^2*x^4+495*a^2*b*x^2+385*a^3)*((b*x^2+a)^2)^(3/2)*(d
*x)^(1/2)/(b*x^2+a)^3

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Maxima [A]  time = 0.709566, size = 112, normalized size = 0.57 \[ \frac{2}{165} \,{\left (11 \, b^{3} \sqrt{d} x^{3} + 15 \, a b^{2} \sqrt{d} x\right )} x^{\frac{9}{2}} + \frac{4}{77} \,{\left (7 \, a b^{2} \sqrt{d} x^{3} + 11 \, a^{2} b \sqrt{d} x\right )} x^{\frac{5}{2}} + \frac{2}{21} \,{\left (3 \, a^{2} b \sqrt{d} x^{3} + 7 \, a^{3} \sqrt{d} x\right )} \sqrt{x} \]

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)*sqrt(d*x),x, algorithm="maxima")

[Out]

2/165*(11*b^3*sqrt(d)*x^3 + 15*a*b^2*sqrt(d)*x)*x^(9/2) + 4/77*(7*a*b^2*sqrt(d)*
x^3 + 11*a^2*b*sqrt(d)*x)*x^(5/2) + 2/21*(3*a^2*b*sqrt(d)*x^3 + 7*a^3*sqrt(d)*x)
*sqrt(x)

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Fricas [A]  time = 0.270706, size = 54, normalized size = 0.28 \[ \frac{2}{1155} \,{\left (77 \, b^{3} x^{7} + 315 \, a b^{2} x^{5} + 495 \, a^{2} b x^{3} + 385 \, a^{3} x\right )} \sqrt{d x} \]

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)*sqrt(d*x),x, algorithm="fricas")

[Out]

2/1155*(77*b^3*x^7 + 315*a*b^2*x^5 + 495*a^2*b*x^3 + 385*a^3*x)*sqrt(d*x)

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

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)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.266292, size = 127, normalized size = 0.65 \[ \frac{2 \,{\left (77 \, \sqrt{d x} b^{3} d x^{7}{\rm sign}\left (b x^{2} + a\right ) + 315 \, \sqrt{d x} a b^{2} d x^{5}{\rm sign}\left (b x^{2} + a\right ) + 495 \, \sqrt{d x} a^{2} b d x^{3}{\rm sign}\left (b x^{2} + a\right ) + 385 \, \sqrt{d x} a^{3} d x{\rm sign}\left (b x^{2} + a\right )\right )}}{1155 \, d} \]

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)*sqrt(d*x),x, algorithm="giac")

[Out]

2/1155*(77*sqrt(d*x)*b^3*d*x^7*sign(b*x^2 + a) + 315*sqrt(d*x)*a*b^2*d*x^5*sign(
b*x^2 + a) + 495*sqrt(d*x)*a^2*b*d*x^3*sign(b*x^2 + a) + 385*sqrt(d*x)*a^3*d*x*s
ign(b*x^2 + a))/d