3.598 \(\int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=329 \[ -\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]

[Out]

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

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Rubi [A]  time = 0.670314, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(5/2)*(a + b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 68.9929, size = 303, normalized size = 0.92 \[ \frac{8 a^{\frac{13}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{4 a^{\frac{13}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{8 a^{3} c^{2} \sqrt{c x} \sqrt{a + b x^{2}}}{65 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{8 a^{2} c \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{195 b} + \frac{4 a \left (c x\right )^{\frac{7}{2}} \sqrt{a + b x^{2}}}{39 c} + \frac{2 \left (c x\right )^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}}}{13 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(5/2)*(b*x**2+a)**(3/2),x)

[Out]

8*a**(13/4)*c**(5/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt
(b)*x)*elliptic_e(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqrt(c))), 1/2)/(65*b**(7/
4)*sqrt(a + b*x**2)) - 4*a**(13/4)*c**(5/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)
*x)**2)*(sqrt(a) + sqrt(b)*x)*elliptic_f(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqr
t(c))), 1/2)/(65*b**(7/4)*sqrt(a + b*x**2)) - 8*a**3*c**2*sqrt(c*x)*sqrt(a + b*x
**2)/(65*b**(3/2)*(sqrt(a) + sqrt(b)*x)) + 8*a**2*c*(c*x)**(3/2)*sqrt(a + b*x**2
)/(195*b) + 4*a*(c*x)**(7/2)*sqrt(a + b*x**2)/(39*c) + 2*(c*x)**(7/2)*(a + b*x**
2)**(3/2)/(13*c)

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Mathematica [C]  time = 0.310453, size = 202, normalized size = 0.61 \[ \frac{2 c^2 \sqrt{c x} \left (12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (4 a^3+29 a^2 b x^2+40 a b^2 x^4+15 b^3 x^6\right )\right )}{195 b^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(5/2)*(a + b*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.033, size = 232, normalized size = 0.7 \[ -{\frac{2\,{c}^{2}}{195\,{b}^{2}x}\sqrt{cx} \left ( -15\,{x}^{8}{b}^{4}-40\,{x}^{6}a{b}^{3}+12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{4}-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{4}-29\,{x}^{4}{a}^{2}{b}^{2}-4\,{x}^{2}{a}^{3}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(5/2)*(b*x^2+a)^(3/2),x)

[Out]

-2/195*c^2/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b^2*(-15*x^8*b^4-40*x^6*a*b^3+12*((b*x+
(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b
/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/
2))*2^(1/2)*a^4-6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(
-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b
)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^4-29*x^4*a^2*b^2-4*x^2*a^3*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(3/2)*(c*x)^(5/2), x)