3.2960 \(\int x^3 \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=843 \[ \frac{4}{19} \sqrt{a+b \sqrt{c x^3}} x^4+\frac{12 a \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}} x}{247 b c}-\frac{120 a^2 \sqrt{a+b \sqrt{c x^3}} x}{1729 b^2 c}-\frac{240 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{160 \sqrt{2} 3^{3/4} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{480 a^3 \sqrt{a+b \sqrt{c x^3}}}{1729 b^{8/3} c^{4/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )} \]

[Out]

(-120*a^2*x*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^2*c) + (4*x^4*Sqrt[a + b*Sqrt[c*x^3
]])/19 + (12*a*x*Sqrt[c*x^3]*Sqrt[a + b*Sqrt[c*x^3]])/(247*b*c) + (480*a^3*Sqrt[
a + b*Sqrt[c*x^3]])/(1729*b^(8/3)*c^(4/3)*((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2
/3)*x^2)/Sqrt[c*x^3])) - (240*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + (b^(
1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1
/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*S
qrt[3]])/(1729*b^(8/3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sq
rt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[
a + b*Sqrt[c*x^3]]) + (160*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*
x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x
^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1
+ Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(172
9*b^(8/3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/(
(1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*
x^3]])

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Rubi [A]  time = 1.50679, antiderivative size = 843, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{4}{19} \sqrt{a+b \sqrt{c x^3}} x^4+\frac{12 a \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}} x}{247 b c}-\frac{120 a^2 \sqrt{a+b \sqrt{c x^3}} x}{1729 b^2 c}-\frac{240 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{160 \sqrt{2} 3^{3/4} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{480 a^3 \sqrt{a+b \sqrt{c x^3}}}{1729 b^{8/3} c^{4/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-120*a^2*x*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^2*c) + (4*x^4*Sqrt[a + b*Sqrt[c*x^3
]])/19 + (12*a*x*Sqrt[c*x^3]*Sqrt[a + b*Sqrt[c*x^3]])/(247*b*c) + (480*a^3*Sqrt[
a + b*Sqrt[c*x^3]])/(1729*b^(8/3)*c^(4/3)*((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2
/3)*x^2)/Sqrt[c*x^3])) - (240*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + (b^(
1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1
/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*S
qrt[3]])/(1729*b^(8/3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sq
rt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[
a + b*Sqrt[c*x^3]]) + (160*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*
x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x
^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1
+ Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(172
9*b^(8/3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/(
(1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*
x^3]])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0420275, size = 0, normalized size = 0. \[ \int x^3 \sqrt{a+b \sqrt{c x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.177, size = 932, normalized size = 1.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/1729/c^2/x^2*(30*I*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b
^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*((b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3
)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)+2
*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*EllipticE
(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b
^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))
^(1/2))*3^(1/2)*x^2*2^(1/2)*(-a*c*b^2)^(2/3)*a^3-20*I*(-I*(I*3^(1/2)*x*(-a*c*b^2
)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*
((b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-
I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-
a*c*b^2)^(1/3)/x)^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2
)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2),
2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^2*2^(1/2)*(-a*c*b^2)^(2/3)*a^
3+91*(c*x^3)^(1/2)*x^6*b^5*c^2+112*x^6*a*b^4*c^2+30*(-I*(I*3^(1/2)*x*(-a*c*b^2)^
(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*((
b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*
(I*3^(1/2)*x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*
c*b^2)^(1/3)/x)^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^
(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2),2^
(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*x^2*2^(1/2)*(-a*c*b^2)^(2/3)*a^3-30*(c*x^
3)^(1/2)*x^3*a^2*b^3*c-30*x^3*a^3*b^2*c+21*(c*x^3)^(3/2)*a^2*b^3)/b^4/(a+b*(c*x^
3)^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{3}} b + a} x^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(sqrt(c*x^3)*b + a)*x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{c x^{3}} b + a} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*x^3, x)