3.232 \(\int x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=321 \[ \frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]

[Out]

(-24*b^4*(13*b*B - 23*A*c)*Sqrt[b*x^2 + c*x^4])/(33649*c^4*Sqrt[x]) + (72*b^3*(1
3*b*B - 23*A*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(168245*c^3) - (8*b^2*(13*b*B - 23*
A*c)*x^(7/2)*Sqrt[b*x^2 + c*x^4])/(24035*c^2) - (4*b*(13*b*B - 23*A*c)*x^(11/2)*
Sqrt[b*x^2 + c*x^4])/(2185*c) - (2*(13*b*B - 23*A*c)*x^(7/2)*(b*x^2 + c*x^4)^(3/
2))/(437*c) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(5/2))/(23*c) + (12*b^(19/4)*(13*b*B
- 23*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(33649*c^(17/4)*Sqrt[b*x^2 + c*
x^4])

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Rubi [A]  time = 0.870581, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(-24*b^4*(13*b*B - 23*A*c)*Sqrt[b*x^2 + c*x^4])/(33649*c^4*Sqrt[x]) + (72*b^3*(1
3*b*B - 23*A*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(168245*c^3) - (8*b^2*(13*b*B - 23*
A*c)*x^(7/2)*Sqrt[b*x^2 + c*x^4])/(24035*c^2) - (4*b*(13*b*B - 23*A*c)*x^(11/2)*
Sqrt[b*x^2 + c*x^4])/(2185*c) - (2*(13*b*B - 23*A*c)*x^(7/2)*(b*x^2 + c*x^4)^(3/
2))/(437*c) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(5/2))/(23*c) + (12*b^(19/4)*(13*b*B
- 23*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(33649*c^(17/4)*Sqrt[b*x^2 + c*
x^4])

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Rubi in Sympy [A]  time = 74.5547, size = 314, normalized size = 0.98 \[ \frac{2 B x^{\frac{3}{2}} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{23 c} - \frac{12 b^{\frac{19}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (23 A c - 13 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{33649 c^{\frac{17}{4}} x \left (b + c x^{2}\right )} + \frac{24 b^{4} \left (23 A c - 13 B b\right ) \sqrt{b x^{2} + c x^{4}}}{33649 c^{4} \sqrt{x}} - \frac{72 b^{3} x^{\frac{3}{2}} \left (23 A c - 13 B b\right ) \sqrt{b x^{2} + c x^{4}}}{168245 c^{3}} + \frac{8 b^{2} x^{\frac{7}{2}} \left (23 A c - 13 B b\right ) \sqrt{b x^{2} + c x^{4}}}{24035 c^{2}} + \frac{4 b x^{\frac{11}{2}} \left (23 A c - 13 B b\right ) \sqrt{b x^{2} + c x^{4}}}{2185 c} + \frac{2 x^{\frac{7}{2}} \left (23 A c - 13 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{437 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

2*B*x**(3/2)*(b*x**2 + c*x**4)**(5/2)/(23*c) - 12*b**(19/4)*sqrt((b + c*x**2)/(s
qrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(23*A*c - 13*B*b)*sqrt(b*x**2 + c*
x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(33649*c**(17/4)*x*(b +
c*x**2)) + 24*b**4*(23*A*c - 13*B*b)*sqrt(b*x**2 + c*x**4)/(33649*c**4*sqrt(x))
- 72*b**3*x**(3/2)*(23*A*c - 13*B*b)*sqrt(b*x**2 + c*x**4)/(168245*c**3) + 8*b*
*2*x**(7/2)*(23*A*c - 13*B*b)*sqrt(b*x**2 + c*x**4)/(24035*c**2) + 4*b*x**(11/2)
*(23*A*c - 13*B*b)*sqrt(b*x**2 + c*x**4)/(2185*c) + 2*x**(7/2)*(23*A*c - 13*B*b)
*(b*x**2 + c*x**4)**(3/2)/(437*c)

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Mathematica [C]  time = 0.770537, size = 219, normalized size = 0.68 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{12 b^4 c \left (115 A+39 B x^2\right )-4 b^3 c^2 x^2 \left (207 A+91 B x^2\right )+28 b^2 c^3 x^4 \left (23 A+11 B x^2\right )+77 b c^4 x^6 \left (161 A+125 B x^2\right )+385 c^5 x^8 \left (23 A+19 B x^2\right )-780 b^5 B}{\sqrt{x}}+\frac{60 i b^5 \sqrt{\frac{b}{c x^2}+1} (13 b B-23 A c) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right )}{168245 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(2*Sqrt[x^2*(b + c*x^2)]*((-780*b^5*B + 28*b^2*c^3*x^4*(23*A + 11*B*x^2) + 385*c
^5*x^8*(23*A + 19*B*x^2) + 12*b^4*c*(115*A + 39*B*x^2) - 4*b^3*c^2*x^2*(207*A +
91*B*x^2) + 77*b*c^4*x^6*(161*A + 125*B*x^2))/Sqrt[x] + ((60*I)*b^5*(13*b*B - 23
*A*c)*Sqrt[1 + b/(c*x^2)]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]]
, -1])/(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2))))/(168245*c^4)

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Maple [A]  time = 0.053, size = 355, normalized size = 1.1 \[ -{\frac{2}{168245\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -7315\,B{x}^{13}{c}^{7}-8855\,A{x}^{11}{c}^{7}-16940\,B{x}^{11}b{c}^{6}-21252\,A{x}^{9}b{c}^{6}-9933\,B{x}^{9}{b}^{2}{c}^{5}-13041\,A{x}^{7}{b}^{2}{c}^{5}+56\,B{x}^{7}{b}^{3}{c}^{4}+690\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{5}c+184\,A{x}^{5}{b}^{3}{c}^{4}-390\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{6}-104\,B{x}^{5}{b}^{4}{c}^{3}-552\,A{x}^{3}{b}^{4}{c}^{3}+312\,B{x}^{3}{b}^{5}{c}^{2}-1380\,Ax{b}^{5}{c}^{2}+780\,Bx{b}^{6}c \right ){x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)

[Out]

-2/168245*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2*(-7315*B*x^13*c^7-8855*A*x^11*
c^7-16940*B*x^11*b*c^6-21252*A*x^9*b*c^6-9933*B*x^9*b^2*c^5-13041*A*x^7*b^2*c^5+
56*B*x^7*b^3*c^4+690*A*((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))*(-b*c)^(1/2)*b^5*c+184*A*x^5*b^3*c^4-39
0*B*((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))*(-b*c)^(1/2)*b^6-104*B*x^5*b^4*c^3-552*A*x^3*b^4*c^3+312*B
*x^3*b^5*c^2-1380*A*x*b^5*c^2+780*B*x*b^6*c)/c^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B c x^{8} +{\left (B b + A c\right )} x^{6} + A b x^{4}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2),x, algorithm="fricas")

[Out]

integral((B*c*x^8 + (B*b + A*c)*x^6 + A*b*x^4)*sqrt(c*x^4 + b*x^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(x**(5/2)*(B*x**2+A)*(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. \[ \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2), x)