3.377 \(\int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{23/2}} \, dx\)

Optimal. Leaf size=203 \[ \frac{4 c^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]

[Out]

(-4*c*Sqrt[b*x^2 + c*x^4])/(55*x^(13/2)) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(385*b*x^
(9/2)) + (8*c^3*Sqrt[b*x^2 + c*x^4])/(231*b^2*x^(5/2)) - (2*(b*x^2 + c*x^4)^(3/2
))/(15*x^(21/2)) + (4*c^(15/4)*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])/(231*b^(9/
4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.551647, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{4 c^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2)/x^(23/2),x]

[Out]

(-4*c*Sqrt[b*x^2 + c*x^4])/(55*x^(13/2)) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(385*b*x^
(9/2)) + (8*c^3*Sqrt[b*x^2 + c*x^4])/(231*b^2*x^(5/2)) - (2*(b*x^2 + c*x^4)^(3/2
))/(15*x^(21/2)) + (4*c^(15/4)*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])/(231*b^(9/
4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 59.3392, size = 194, normalized size = 0.96 \[ - \frac{4 c \sqrt{b x^{2} + c x^{4}}}{55 x^{\frac{13}{2}}} - \frac{2 \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{15 x^{\frac{21}{2}}} - \frac{8 c^{2} \sqrt{b x^{2} + c x^{4}}}{385 b x^{\frac{9}{2}}} + \frac{8 c^{3} \sqrt{b x^{2} + c x^{4}}}{231 b^{2} x^{\frac{5}{2}}} + \frac{4 c^{\frac{15}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**(23/2),x)

[Out]

-4*c*sqrt(b*x**2 + c*x**4)/(55*x**(13/2)) - 2*(b*x**2 + c*x**4)**(3/2)/(15*x**(2
1/2)) - 8*c**2*sqrt(b*x**2 + c*x**4)/(385*b*x**(9/2)) + 8*c**3*sqrt(b*x**2 + c*x
**4)/(231*b**2*x**(5/2)) + 4*c**(15/4)*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)/(231*b**(9/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.350614, size = 165, normalized size = 0.81 \[ \frac{2 \left (\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (-77 b^4-196 b^3 c x^2-131 b^2 c^2 x^4+8 b c^3 x^6+20 c^4 x^8\right )+20 i c^4 x^{17/2} \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{1155 b^2 x^{13/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(3/2)/x^(23/2),x]

[Out]

(2*(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(-77*b^4 - 196*b^3*c*x^2 - 131*b^2*c^2*x^4 + 8*b*c
^3*x^6 + 20*c^4*x^8) + (20*I)*c^4*Sqrt[1 + b/(c*x^2)]*x^(17/2)*EllipticF[I*ArcSi
nh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1]))/(1155*b^2*Sqrt[(I*Sqrt[b])/Sqrt[c]]
*x^(13/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.045, size = 167, normalized size = 0.8 \[{\frac{2}{1155\, \left ( c{x}^{2}+b \right ) ^{2}{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 10\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}\sqrt{2}{x}^{7}{c}^{3}+20\,{x}^{8}{c}^{4}+8\,{x}^{6}b{c}^{3}-131\,{x}^{4}{b}^{2}{c}^{2}-196\,{x}^{2}{b}^{3}c-77\,{b}^{4} \right ){x}^{-{\frac{21}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^(23/2),x)

[Out]

2/1155*(c*x^4+b*x^2)^(3/2)/x^(21/2)/(c*x^2+b)^2*(10*((c*x+(-b*c)^(1/2))/(-b*c)^(
1/2))^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*E
llipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*2^(1/
2)*x^7*c^3+20*x^8*c^4+8*x^6*b*c^3-131*x^4*b^2*c^2-196*x^2*b^3*c-77*b^4)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{23}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^(23/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)/x^(23/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{x^{\frac{19}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^(23/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)*(c*x^2 + b)/x^(19/2), 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**4+b*x**2)**(3/2)/x**(23/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{23}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^(23/2),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)/x^(23/2), x)