3.245 \(\int \frac{x^{11/2} \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=369 \[ -\frac{7 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{14 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{585 c^3}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c} \]

[Out]

(-14*b^2*(11*b*B - 13*A*c)*x^(3/2)*(b + c*x^2))/(195*c^(7/2)*(Sqrt[b] + Sqrt[c]*
x)*Sqrt[b*x^2 + c*x^4]) + (14*b*(11*b*B - 13*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(
585*c^3) - (2*(11*b*B - 13*A*c)*x^(5/2)*Sqrt[b*x^2 + c*x^4])/(117*c^2) + (2*B*x^
(9/2)*Sqrt[b*x^2 + c*x^4])/(13*c) + (14*b^(9/4)*(11*b*B - 13*A*c)*x*(Sqrt[b] + S
qrt[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])/(195*c^(15/4)*Sqrt[b*x^2 + c*x^4]) - (7*b^(9/4)*(11*b*B
- 13*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ell
ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(195*c^(15/4)*Sqrt[b*x^2 + c*x
^4])

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Rubi [A]  time = 0.835888, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{7 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{14 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{585 c^3}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(11/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(-14*b^2*(11*b*B - 13*A*c)*x^(3/2)*(b + c*x^2))/(195*c^(7/2)*(Sqrt[b] + Sqrt[c]*
x)*Sqrt[b*x^2 + c*x^4]) + (14*b*(11*b*B - 13*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(
585*c^3) - (2*(11*b*B - 13*A*c)*x^(5/2)*Sqrt[b*x^2 + c*x^4])/(117*c^2) + (2*B*x^
(9/2)*Sqrt[b*x^2 + c*x^4])/(13*c) + (14*b^(9/4)*(11*b*B - 13*A*c)*x*(Sqrt[b] + S
qrt[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])/(195*c^(15/4)*Sqrt[b*x^2 + c*x^4]) - (7*b^(9/4)*(11*b*B
- 13*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ell
ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(195*c^(15/4)*Sqrt[b*x^2 + c*x
^4])

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Rubi in Sympy [A]  time = 71.5959, size = 357, normalized size = 0.97 \[ \frac{2 B x^{\frac{9}{2}} \sqrt{b x^{2} + c x^{4}}}{13 c} - \frac{14 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (13 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{195 c^{\frac{15}{4}} x \left (b + c x^{2}\right )} + \frac{7 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (13 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{195 c^{\frac{15}{4}} x \left (b + c x^{2}\right )} + \frac{14 b^{2} \left (13 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{195 c^{\frac{7}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{14 b \sqrt{x} \left (13 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{585 c^{3}} + \frac{2 x^{\frac{5}{2}} \left (13 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{117 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(11/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

2*B*x**(9/2)*sqrt(b*x**2 + c*x**4)/(13*c) - 14*b**(9/4)*sqrt((b + c*x**2)/(sqrt(
b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(13*A*c - 11*B*b)*sqrt(b*x**2 + c*x**4
)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(195*c**(15/4)*x*(b + c*x**
2)) + 7*b**(9/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*
x)*(13*A*c - 11*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b*
*(1/4)), 1/2)/(195*c**(15/4)*x*(b + c*x**2)) + 14*b**2*(13*A*c - 11*B*b)*sqrt(b*
x**2 + c*x**4)/(195*c**(7/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) - 14*b*sqrt(x)*(13*A
*c - 11*B*b)*sqrt(b*x**2 + c*x**4)/(585*c**3) + 2*x**(5/2)*(13*A*c - 11*B*b)*sqr
t(b*x**2 + c*x**4)/(117*c**2)

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Mathematica [C]  time = 1.01127, size = 264, normalized size = 0.72 \[ \frac{2 x \left (c x^{3/2} \left (b+c x^2\right ) \left (-b c \left (91 A+55 B x^2\right )+5 c^2 x^2 \left (13 A+9 B x^2\right )+77 b^2 B\right )-\frac{21 b^2 (11 b B-13 A c) \left (\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )+\sqrt{b} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{b} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{x} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}\right )}{585 c^4 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(11/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(2*x*(c*x^(3/2)*(b + c*x^2)*(77*b^2*B + 5*c^2*x^2*(13*A + 9*B*x^2) - b*c*(91*A +
55*B*x^2)) - (21*b^2*(11*b*B - 13*A*c)*(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2) -
Sqrt[b]*Sqrt[c]*Sqrt[1 + b/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]
)/Sqrt[c]]/Sqrt[x]], -1] + Sqrt[b]*Sqrt[c]*Sqrt[1 + b/(c*x^2)]*x^(3/2)*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1]))/(Sqrt[(I*Sqrt[b])/Sqrt[c]]*
Sqrt[x])))/(585*c^4*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.042, size = 437, normalized size = 1.2 \[{\frac{1}{585\,{c}^{4}}\sqrt{x} \left ( 90\,B{x}^{8}{c}^{4}+130\,A{x}^{6}{c}^{4}-20\,B{x}^{6}b{c}^{3}+546\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}c-273\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}c-462\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{4}+231\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{4}-52\,A{x}^{4}b{c}^{3}+44\,B{x}^{4}{b}^{2}{c}^{2}-182\,A{x}^{2}{b}^{2}{c}^{2}+154\,B{x}^{2}{b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/585/(c*x^4+b*x^2)^(1/2)*x^(1/2)/c^4*(90*B*x^8*c^4+130*A*x^6*c^4-20*B*x^6*b*c^3
+546*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)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(
1/2))^(1/2),1/2*2^(1/2))*b^3*c-273*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)*Elliptic
F(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b^3*c-462*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)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1
/2))*b^4+231*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^4-52*A*x^4*b*c^3+44*B*x^4*b^2*c^2-182*A*x^2*b
^2*c^2+154*B*x^2*b^3*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(11/2)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(11/2)/sqrt(c*x^4 + b*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{7} + A x^{5}\right )} \sqrt{x}}{\sqrt{c x^{4} + b x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(11/2)/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

integral((B*x^7 + A*x^5)*sqrt(x)/sqrt(c*x^4 + b*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(x**(11/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(11/2)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(11/2)/sqrt(c*x^4 + b*x^2), x)