3.590 \(\int (c x)^{5/2} \sqrt{a+b x^2} \, dx\)

Optimal. Leaf size=301 \[ -\frac{2 a^{9/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a^{9/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^2 c^2 \sqrt{c x} \sqrt{a+b x^2}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (c x)^{7/2} \sqrt{a+b x^2}}{9 c}+\frac{4 a c (c x)^{3/2} \sqrt{a+b x^2}}{45 b} \]

[Out]

(4*a*c*(c*x)^(3/2)*Sqrt[a + b*x^2])/(45*b) + (2*(c*x)^(7/2)*Sqrt[a + b*x^2])/(9*
c) - (4*a^2*c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(15*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) +
(4*a^(9/4)*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])/(15*b^(7/4)*
Sqrt[a + b*x^2]) - (2*a^(9/4)*c^(5/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sq
rt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])],
1/2])/(15*b^(7/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.600756, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{2 a^{9/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a^{9/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^2 c^2 \sqrt{c x} \sqrt{a+b x^2}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (c x)^{7/2} \sqrt{a+b x^2}}{9 c}+\frac{4 a c (c x)^{3/2} \sqrt{a+b x^2}}{45 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*c*(c*x)^(3/2)*Sqrt[a + b*x^2])/(45*b) + (2*(c*x)^(7/2)*Sqrt[a + b*x^2])/(9*
c) - (4*a^2*c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(15*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) +
(4*a^(9/4)*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])/(15*b^(7/4)*
Sqrt[a + b*x^2]) - (2*a^(9/4)*c^(5/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sq
rt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])],
1/2])/(15*b^(7/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 57.923, size = 277, normalized size = 0.92 \[ \frac{4 a^{\frac{9}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{2 a^{\frac{9}{4}} c^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{4 a^{2} c^{2} \sqrt{c x} \sqrt{a + b x^{2}}}{15 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{4 a c \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{45 b} + \frac{2 \left (c x\right )^{\frac{7}{2}} \sqrt{a + b x^{2}}}{9 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(5/2)*(b*x**2+a)**(1/2),x)

[Out]

4*a**(9/4)*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)/(15*b**(7/4
)*sqrt(a + b*x**2)) - 2*a**(9/4)*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)*sqrt(
c))), 1/2)/(15*b**(7/4)*sqrt(a + b*x**2)) - 4*a**2*c**2*sqrt(c*x)*sqrt(a + b*x**
2)/(15*b**(3/2)*(sqrt(a) + sqrt(b)*x)) + 4*a*c*(c*x)**(3/2)*sqrt(a + b*x**2)/(45
*b) + 2*(c*x)**(7/2)*sqrt(a + b*x**2)/(9*c)

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Mathematica [C]  time = 0.342254, size = 191, normalized size = 0.63 \[ \frac{2 c^2 \sqrt{c x} \left (6 a^{5/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-6 a^{5/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (2 a^2+7 a b x^2+5 b^2 x^4\right )\right )}{45 b^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*Sqrt[c*x]*(Sqrt[b]*x*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(2*a^2 + 7*a*b*x^2 + 5*b
^2*x^4) - 6*a^(5/2)*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/S
qrt[a]]], -1] + 6*a^(5/2)*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b
]*x)/Sqrt[a]]], -1]))/(45*b^(3/2)*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.032, size = 221, normalized size = 0.7 \[ -{\frac{2\,{c}^{2}}{45\,{b}^{2}x}\sqrt{cx} \left ( -5\,{b}^{3}{x}^{6}+6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{3}-3\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{3}-7\,a{b}^{2}{x}^{4}-2\,{a}^{2}b{x}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(5/2)*(b*x^2+a)^(1/2),x)

[Out]

-2/45*c^2/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b^2*(-5*b^3*x^6+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)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3-
3*((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^3-7*a*b^2*x^4-2*a^2*b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(b*x^2 + a)*sqrt(c*x)*c^2*x^2, 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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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