3.239 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=356 \[ \frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{4}{15} \sqrt{x} \sqrt{b x^2+c x^4} (9 A c+b B)+\frac{2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac{8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]

[Out]

(8*b*(b*B + 9*A*c)*x^(3/2)*(b + c*x^2))/(15*Sqrt[c]*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b
*x^2 + c*x^4]) + (4*(b*B + 9*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/15 + (2*(b*B + 9*
A*c)*(b*x^2 + c*x^4)^(3/2))/(9*b*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(b*x^(11
/2)) - (8*b^(5/4)*(b*B + 9*A*c)*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])/(15*c^(3/
4)*Sqrt[b*x^2 + c*x^4]) + (4*b^(5/4)*(b*B + 9*A*c)*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])/(15*c^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.824252, antiderivative size = 356, 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 \[ \frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{4}{15} \sqrt{x} \sqrt{b x^2+c x^4} (9 A c+b B)+\frac{2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac{8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(9/2),x]

[Out]

(8*b*(b*B + 9*A*c)*x^(3/2)*(b + c*x^2))/(15*Sqrt[c]*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b
*x^2 + c*x^4]) + (4*(b*B + 9*A*c)*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/15 + (2*(b*B + 9*
A*c)*(b*x^2 + c*x^4)^(3/2))/(9*b*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(b*x^(11
/2)) - (8*b^(5/4)*(b*B + 9*A*c)*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])/(15*c^(3/
4)*Sqrt[b*x^2 + c*x^4]) + (4*b^(5/4)*(b*B + 9*A*c)*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])/(15*c^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 71.7215, size = 340, normalized size = 0.96 \[ - \frac{2 A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{b x^{\frac{11}{2}}} - \frac{8 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (9 A c + B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{4 b^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (9 A c + B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{8 b \left (9 A c + B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 \sqrt{c} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \sqrt{x} \left (\frac{12 A c}{5} + \frac{4 B b}{15}\right ) \sqrt{b x^{2} + c x^{4}} + \frac{2 \left (9 A c + B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{9 b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**(9/2),x)

[Out]

-2*A*(b*x**2 + c*x**4)**(5/2)/(b*x**(11/2)) - 8*b**(5/4)*sqrt((b + c*x**2)/(sqrt
(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(9*A*c + B*b)*sqrt(b*x**2 + c*x**4)*e
lliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(15*c**(3/4)*x*(b + c*x**2)) +
4*b**(5/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(9
*A*c + B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)),
1/2)/(15*c**(3/4)*x*(b + c*x**2)) + 8*b*(9*A*c + B*b)*sqrt(b*x**2 + c*x**4)/(15*
sqrt(c)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) + sqrt(x)*(12*A*c/5 + 4*B*b/15)*sqrt(b*x*
*2 + c*x**4) + 2*(9*A*c + B*b)*(b*x**2 + c*x**4)**(3/2)/(9*b*x**(3/2))

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Mathematica [C]  time = 1.00637, size = 249, normalized size = 0.7 \[ \frac{2 \sqrt{x} \left (12 b^{3/2} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} (9 A c+b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-12 b^{3/2} \sqrt{c} x^{3/2} \sqrt{\frac{b}{c x^2}+1} (9 A c+b B) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (b c \left (63 A+11 B x^2\right )+c^2 x^2 \left (9 A+5 B x^2\right )+12 b^2 B\right )\right )}{45 c \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(9/2),x]

[Out]

(2*Sqrt[x]*(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2)*(12*b^2*B + c^2*x^2*(9*A + 5*B
*x^2) + b*c*(63*A + 11*B*x^2)) - 12*b^(3/2)*Sqrt[c]*(b*B + 9*A*c)*Sqrt[1 + b/(c*
x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1] + 12*b
^(3/2)*Sqrt[c]*(b*B + 9*A*c)*Sqrt[1 + b/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqr
t[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1]))/(45*Sqrt[(I*Sqrt[b])/Sqrt[c]]*c*Sqrt[x^2*
(b + c*x^2)])

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Maple [A]  time = 0.028, size = 429, normalized size = 1.2 \[{\frac{2}{45\, \left ( c{x}^{2}+b \right ) ^{2}c} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 5\,B{c}^{3}{x}^{6}+108\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c-54\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c+12\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}-6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}+9\,A{x}^{4}{c}^{3}+16\,B{x}^{4}b{c}^{2}-36\,A{x}^{2}b{c}^{2}+11\,B{x}^{2}{b}^{2}c-45\,A{b}^{2}c \right ){x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(9/2),x)

[Out]

2/45*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2*(5*B*c^3*x^6+108*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^2*c-54*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^2*c+12*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)*Ell
ipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b^3-6*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^3+9*A*x^4*c^3+16*B*x^4*b*c^2-36*A*x^2*b*c^2+11*B*x^2*b^2*c-45*A*b^2*c)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}\,{d x} \]

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^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^(9/2), x)

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

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^(9/2),x, algorithm="fricas")

[Out]

integral((B*c*x^4 + (B*b + A*c)*x^2 + A*b)*sqrt(c*x^4 + b*x^2)/x^(5/2), 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((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}\,{d x} \]

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^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^(9/2), x)