3.2924 \(\int x^3 \sqrt{a+b \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{2 a^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac{6 a^2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^4 c^2}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac{6 a \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.140097, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 a^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac{6 a^2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^4 c^2}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac{6 a \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

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

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Rubi in Sympy [A]  time = 17.3633, size = 109, normalized size = 0.94 \[ - \frac{2 a^{3} \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b^{4} c^{2}} + \frac{6 a^{2} \left (a + b \sqrt{c x^{2}}\right )^{\frac{5}{2}}}{5 b^{4} c^{2}} - \frac{6 a \left (a + b \sqrt{c x^{2}}\right )^{\frac{7}{2}}}{7 b^{4} c^{2}} + \frac{2 \left (a + b \sqrt{c x^{2}}\right )^{\frac{9}{2}}}{9 b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

-2*a**3*(a + b*sqrt(c*x**2))**(3/2)/(3*b**4*c**2) + 6*a**2*(a + b*sqrt(c*x**2))*
*(5/2)/(5*b**4*c**2) - 6*a*(a + b*sqrt(c*x**2))**(7/2)/(7*b**4*c**2) + 2*(a + b*
sqrt(c*x**2))**(9/2)/(9*b**4*c**2)

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Mathematica [A]  time = 0.0456117, size = 86, normalized size = 0.74 \[ \frac{2 \sqrt{a+b \sqrt{c x^2}} \left (-16 a^4+8 a^3 b \sqrt{c x^2}-6 a^2 b^2 c x^2+5 a b^3 \left (c x^2\right )^{3/2}+35 b^4 c^2 x^4\right )}{315 b^4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*Sqrt[a + b*Sqrt[c*x^2]]*(-16*a^4 - 6*a^2*b^2*c*x^2 + 35*b^4*c^2*x^4 + 8*a^3*b
*Sqrt[c*x^2] + 5*a*b^3*(c*x^2)^(3/2)))/(315*b^4*c^2)

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Maple [A]  time = 0.008, size = 63, normalized size = 0.5 \[{\frac{2}{315\,{c}^{2}{b}^{4}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( 35\, \left ( c{x}^{2} \right ) ^{3/2}{b}^{3}-30\,c{x}^{2}a{b}^{2}+24\,\sqrt{c{x}^{2}}{a}^{2}b-16\,{a}^{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*(c*x^2)^(1/2))^(1/2),x)

[Out]

2/315*(a+b*(c*x^2)^(1/2))^(3/2)*(35*(c*x^2)^(3/2)*b^3-30*c*x^2*a*b^2+24*(c*x^2)^
(1/2)*a^2*b-16*a^3)/c^2/b^4

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Maxima [A]  time = 1.37797, size = 115, normalized size = 0.99 \[ \frac{2 \,{\left (\frac{35 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^3,x, algorithm="maxima")

[Out]

2/315*(35*(sqrt(c*x^2)*b + a)^(9/2)/b^4 - 135*(sqrt(c*x^2)*b + a)^(7/2)*a/b^4 +
189*(sqrt(c*x^2)*b + a)^(5/2)*a^2/b^4 - 105*(sqrt(c*x^2)*b + a)^(3/2)*a^3/b^4)/c
^2

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Fricas [A]  time = 0.208497, size = 101, normalized size = 0.87 \[ \frac{2 \,{\left (35 \, b^{4} c^{2} x^{4} - 6 \, a^{2} b^{2} c x^{2} - 16 \, a^{4} +{\left (5 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{315 \, b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^3,x, algorithm="fricas")

[Out]

2/315*(35*b^4*c^2*x^4 - 6*a^2*b^2*c*x^2 - 16*a^4 + (5*a*b^3*c*x^2 + 8*a^3*b)*sqr
t(c*x^2))*sqrt(sqrt(c*x^2)*b + a)/(b^4*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \sqrt{a + b \sqrt{c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220537, size = 119, normalized size = 1.03 \[ \frac{2 \,{\left (35 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} b^{24} c^{\frac{33}{2}} - 135 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a b^{24} c^{\frac{33}{2}} + 189 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{2} b^{24} c^{\frac{33}{2}} - 105 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{3} b^{24} c^{\frac{33}{2}}\right )}}{315 \, b^{28} c^{\frac{37}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^3,x, algorithm="giac")

[Out]

2/315*(35*(b*sqrt(c)*x + a)^(9/2)*b^24*c^(33/2) - 135*(b*sqrt(c)*x + a)^(7/2)*a*
b^24*c^(33/2) + 189*(b*sqrt(c)*x + a)^(5/2)*a^2*b^24*c^(33/2) - 105*(b*sqrt(c)*x
+ a)^(3/2)*a^3*b^24*c^(33/2))/(b^28*c^(37/2))