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

Optimal. Leaf size=320 \[ -\frac{4 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{8 b^3 x^{3/2} \left (b+c x^2\right )}{65 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}{195 c}+\frac{2}{13} \sqrt{x} \left (b x^2+c x^4\right )^{3/2}+\frac{4}{39} b x^{5/2} \sqrt{b x^2+c x^4} \]

[Out]

(-8*b^3*x^(3/2)*(b + c*x^2))/(65*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^
4]) + (8*b^2*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(195*c) + (4*b*x^(5/2)*Sqrt[b*x^2 + c*
x^4])/39 + (2*Sqrt[x]*(b*x^2 + c*x^4)^(3/2))/13 + (8*b^(13/4)*x*(Sqrt[b] + Sqrt[
c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt
[x])/b^(1/4)], 1/2])/(65*c^(7/4)*Sqrt[b*x^2 + c*x^4]) - (4*b^(13/4)*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])/(65*c^(7/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.722549, antiderivative size = 320, 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 \[ -\frac{4 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{8 b^3 x^{3/2} \left (b+c x^2\right )}{65 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}{195 c}+\frac{2}{13} \sqrt{x} \left (b x^2+c x^4\right )^{3/2}+\frac{4}{39} b x^{5/2} \sqrt{b x^2+c x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-8*b^3*x^(3/2)*(b + c*x^2))/(65*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^
4]) + (8*b^2*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(195*c) + (4*b*x^(5/2)*Sqrt[b*x^2 + c*
x^4])/39 + (2*Sqrt[x]*(b*x^2 + c*x^4)^(3/2))/13 + (8*b^(13/4)*x*(Sqrt[b] + Sqrt[
c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt
[x])/b^(1/4)], 1/2])/(65*c^(7/4)*Sqrt[b*x^2 + c*x^4]) - (4*b^(13/4)*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])/(65*c^(7/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 67.419, size = 301, normalized size = 0.94 \[ \frac{8 b^{\frac{13}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{65 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{4 b^{\frac{13}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{65 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{8 b^{3} \sqrt{b x^{2} + c x^{4}}}{65 c^{\frac{3}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{8 b^{2} \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{195 c} + \frac{4 b x^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{39} + \frac{2 \sqrt{x} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**(1/2),x)

[Out]

8*b**(13/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sq
rt(b*x**2 + c*x**4)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(65*c**(7
/4)*x*(b + c*x**2)) - 4*b**(13/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(s
qrt(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)/(65*c**(7/4)*x*(b + c*x**2)) - 8*b**3*sqrt(b*x**2 + c*x**4)/(65*c**
(3/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) + 8*b**2*sqrt(x)*sqrt(b*x**2 + c*x**4)/(195
*c) + 4*b*x**(5/2)*sqrt(b*x**2 + c*x**4)/39 + 2*sqrt(x)*(b*x**2 + c*x**4)**(3/2)
/13

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Mathematica [C]  time = 0.325854, size = 201, normalized size = 0.63 \[ \frac{2 x^{3/2} \left (12 b^{7/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-12 b^{7/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (4 b^3+29 b^2 c x^2+40 b c^2 x^4+15 c^3 x^6\right )\right )}{195 c^{3/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(Sqrt[c]*x*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(4*b^3 + 29*b^2*c*x^2 + 40*b*c
^2*x^4 + 15*c^3*x^6) - 12*b^(7/2)*Sqrt[1 + (c*x^2)/b]*EllipticE[I*ArcSinh[Sqrt[(
I*Sqrt[c]*x)/Sqrt[b]]], -1] + 12*b^(7/2)*Sqrt[1 + (c*x^2)/b]*EllipticF[I*ArcSinh
[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1]))/(195*c^(3/2)*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*Sq
rt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.018, size = 237, normalized size = 0.7 \[ -{\frac{2}{195\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -15\,{x}^{8}{c}^{4}-40\,{x}^{6}b{c}^{3}+12\,{b}^{4}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -6\,{b}^{4}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -29\,{x}^{4}{b}^{2}{c}^{2}-4\,{x}^{2}{b}^{3}c \right ){x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^(1/2),x)

[Out]

-2/195*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2/c^2*(-15*x^8*c^4-40*x^6*b*c^3+12*
b^4*((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))-6*b^4*((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))-29*x^4*b^2*c^2-4*x^2*b^3*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^4 + b*x^2)^(3/2)/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((c*x**4+b*x**2)**(3/2)/x**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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