3.392 \(\int \frac{x^{15/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 b x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{7 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c^2}-\frac{x^{9/2}}{c \sqrt{b x^2+c x^4}} \]

[Out]

-(x^(9/2)/(c*Sqrt[b*x^2 + c*x^4])) - (21*b*x^(3/2)*(b + c*x^2))/(5*c^(5/2)*(Sqrt
[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) + (7*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(5*c^2)
+ (21*b^(5/4)*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])/(5*c^(11/4)*Sqrt[b*x^2 + c*
x^4]) - (21*b^(5/4)*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])/(10*c^(11/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi [A]  time = 0.624546, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 b x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{7 \sqrt{x} \sqrt{b x^2+c x^4}}{5 c^2}-\frac{x^{9/2}}{c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^(15/2)/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(x^(9/2)/(c*Sqrt[b*x^2 + c*x^4])) - (21*b*x^(3/2)*(b + c*x^2))/(5*c^(5/2)*(Sqrt
[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) + (7*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(5*c^2)
+ (21*b^(5/4)*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])/(5*c^(11/4)*Sqrt[b*x^2 + c*
x^4]) - (21*b^(5/4)*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])/(10*c^(11/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi in Sympy [A]  time = 59.285, size = 272, normalized size = 0.93 \[ \frac{21 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} - \frac{21 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{11}{4}} x \left (b + c x^{2}\right )} - \frac{21 b \sqrt{b x^{2} + c x^{4}}}{5 c^{\frac{5}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{x^{\frac{9}{2}}}{c \sqrt{b x^{2} + c x^{4}}} + \frac{7 \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{5 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(15/2)/(c*x**4+b*x**2)**(3/2),x)

[Out]

21*b**(5/4)*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)/(5*c**(11
/4)*x*(b + c*x**2)) - 21*b**(5/4)*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)/(10*c**(11/4)*x*(b + c*x**2)) - 21*b*sqrt(b*x**2 + c*x**4)/(5*c**(5
/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) - x**(9/2)/(c*sqrt(b*x**2 + c*x**4)) + 7*sqrt
(x)*sqrt(b*x**2 + c*x**4)/(5*c**2)

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Mathematica [C]  time = 0.209121, size = 179, normalized size = 0.62 \[ \frac{x^{3/2} \left (21 b^{3/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-21 b^{3/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (7 b+2 c x^2\right )\right )}{5 c^{5/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(15/2)/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x^(3/2)*(Sqrt[c]*x*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(7*b + 2*c*x^2) - 21*b^(3/2)*Sqr
t[1 + (c*x^2)/b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1] + 21*b^(3
/2)*Sqrt[1 + (c*x^2)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1]))/
(5*c^(5/2)*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.046, size = 213, normalized size = 0.7 \[ -{\frac{c{x}^{2}+b}{10\,{c}^{3}}{x}^{{\frac{5}{2}}} \left ( 42\,{b}^{2}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -21\,{b}^{2}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -4\,{c}^{2}{x}^{4}-14\,bc{x}^{2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)/(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/10/(c*x^4+b*x^2)^(3/2)*x^(5/2)*(c*x^2+b)*(42*b^2*((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))-21*b^2*((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))-4*c^2*x^4-14*b*c*x^2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{15}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^(11/2)/(sqrt(c*x^4 + b*x^2)*(c*x^2 + b)), 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**(15/2)/(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. \[ \int \frac{x^{\frac{15}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^(15/2)/(c*x^4 + b*x^2)^(3/2), x)