3.364 \(\int x^{3/2} \left (b x^2+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=350 \[ \frac{28 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{56 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{56 b^4 x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2} \]

[Out]

(56*b^4*x^(3/2)*(b + c*x^2))/(1105*c^(5/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*
x^4]) - (56*b^3*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(3315*c^2) + (8*b^2*x^(5/2)*Sqrt[b*
x^2 + c*x^4])/(663*c) + (12*b*x^(9/2)*Sqrt[b*x^2 + c*x^4])/221 + (2*x^(5/2)*(b*x
^2 + c*x^4)^(3/2))/17 - (56*b^(17/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(S
qrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(110
5*c^(11/4)*Sqrt[b*x^2 + c*x^4]) + (28*b^(17/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b +
c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)],
1/2])/(1105*c^(11/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.865078, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{28 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{56 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{56 b^4 x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(56*b^4*x^(3/2)*(b + c*x^2))/(1105*c^(5/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*
x^4]) - (56*b^3*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(3315*c^2) + (8*b^2*x^(5/2)*Sqrt[b*
x^2 + c*x^4])/(663*c) + (12*b*x^(9/2)*Sqrt[b*x^2 + c*x^4])/221 + (2*x^(5/2)*(b*x
^2 + c*x^4)^(3/2))/17 - (56*b^(17/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(S
qrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(110
5*c^(11/4)*Sqrt[b*x^2 + c*x^4]) + (28*b^(17/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b +
c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)],
1/2])/(1105*c^(11/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 81.9083, size = 330, normalized size = 0.94 \[ - \frac{56 b^{\frac{17}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} + \frac{28 b^{\frac{17}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} + \frac{56 b^{4} \sqrt{b x^{2} + c x^{4}}}{1105 c^{\frac{5}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{56 b^{3} \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{3315 c^{2}} + \frac{8 b^{2} x^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{663 c} + \frac{12 b x^{\frac{9}{2}} \sqrt{b x^{2} + c x^{4}}}{221} + \frac{2 x^{\frac{5}{2}} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(c*x**4+b*x**2)**(3/2),x)

[Out]

-56*b**(17/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*
sqrt(b*x**2 + c*x**4)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(1105*c
**(11/4)*x*(b + c*x**2)) + 28*b**(17/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)*
*2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(
x)/b**(1/4)), 1/2)/(1105*c**(11/4)*x*(b + c*x**2)) + 56*b**4*sqrt(b*x**2 + c*x**
4)/(1105*c**(5/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) - 56*b**3*sqrt(x)*sqrt(b*x**2 +
c*x**4)/(3315*c**2) + 8*b**2*x**(5/2)*sqrt(b*x**2 + c*x**4)/(663*c) + 12*b*x**(
9/2)*sqrt(b*x**2 + c*x**4)/221 + 2*x**(5/2)*(b*x**2 + c*x**4)**(3/2)/17

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Mathematica [C]  time = 0.4362, size = 212, normalized size = 0.61 \[ \frac{2 x^{3/2} \left (-84 b^{9/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+84 b^{9/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (-28 b^4-8 b^3 c x^2+305 b^2 c^2 x^4+480 b c^3 x^6+195 c^4 x^8\right )\right )}{3315 c^{5/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(Sqrt[c]*x*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(-28*b^4 - 8*b^3*c*x^2 + 305*b
^2*c^2*x^4 + 480*b*c^3*x^6 + 195*c^4*x^8) + 84*b^(9/2)*Sqrt[1 + (c*x^2)/b]*Ellip
ticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1] - 84*b^(9/2)*Sqrt[1 + (c*x^2)/b
]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1]))/(3315*c^(5/2)*Sqrt[(I*
Sqrt[c]*x)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.034, size = 248, normalized size = 0.7 \[{\frac{2}{3315\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 195\,{x}^{10}{c}^{5}+480\,{x}^{8}b{c}^{4}+305\,{x}^{6}{b}^{2}{c}^{3}+84\,{b}^{5}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -42\,{b}^{5}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -8\,{x}^{4}{b}^{3}{c}^{2}-28\,{x}^{2}{b}^{4}c \right ){x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3315*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2/c^3*(195*x^10*c^5+480*x^8*b*c^4+3
05*x^6*b^2*c^3+84*b^5*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b
*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticE(((c*x+(-b*c)^
(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))-42*b^5*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))
^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2
)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))-8*x^4*b^3*c^2-2
8*x^2*b^4*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*x^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{5} + b x^{3}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^5 + b*x^3)*sqrt(c*x^4 + b*x^2)*sqrt(x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(c*x**4+b*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*x^(3/2), x)