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

Optimal. Leaf size=282 \[ -\frac{4 b^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-19 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4389 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{b x^2+c x^4} (9 b B-19 A c)}{4389 c^3 \sqrt{x}}-\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{7315 c^2}-\frac{4 b x^{7/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{1045 c}-\frac{2 x^{3/2} \left (b x^2+c x^4\right )^{3/2} (9 b B-19 A c)}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}} \]

[Out]

(8*b^3*(9*b*B - 19*A*c)*Sqrt[b*x^2 + c*x^4])/(4389*c^3*Sqrt[x]) - (8*b^2*(9*b*B
- 19*A*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(7315*c^2) - (4*b*(9*b*B - 19*A*c)*x^(7/2
)*Sqrt[b*x^2 + c*x^4])/(1045*c) - (2*(9*b*B - 19*A*c)*x^(3/2)*(b*x^2 + c*x^4)^(3
/2))/(285*c) + (2*B*(b*x^2 + c*x^4)^(5/2))/(19*c*Sqrt[x]) - (4*b^(15/4)*(9*b*B -
19*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4389*c^(13/4)*Sqrt[b*x^2 + c*x^
4])

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Rubi [A]  time = 0.735483, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{4 b^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-19 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4389 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{b x^2+c x^4} (9 b B-19 A c)}{4389 c^3 \sqrt{x}}-\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{7315 c^2}-\frac{4 b x^{7/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{1045 c}-\frac{2 x^{3/2} \left (b x^2+c x^4\right )^{3/2} (9 b B-19 A c)}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(8*b^3*(9*b*B - 19*A*c)*Sqrt[b*x^2 + c*x^4])/(4389*c^3*Sqrt[x]) - (8*b^2*(9*b*B
- 19*A*c)*x^(3/2)*Sqrt[b*x^2 + c*x^4])/(7315*c^2) - (4*b*(9*b*B - 19*A*c)*x^(7/2
)*Sqrt[b*x^2 + c*x^4])/(1045*c) - (2*(9*b*B - 19*A*c)*x^(3/2)*(b*x^2 + c*x^4)^(3
/2))/(285*c) + (2*B*(b*x^2 + c*x^4)^(5/2))/(19*c*Sqrt[x]) - (4*b^(15/4)*(9*b*B -
19*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4389*c^(13/4)*Sqrt[b*x^2 + c*x^
4])

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Rubi in Sympy [A]  time = 64.0533, size = 275, normalized size = 0.98 \[ \frac{2 B \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{19 c \sqrt{x}} + \frac{4 b^{\frac{15}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (19 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{4389 c^{\frac{13}{4}} x \left (b + c x^{2}\right )} - \frac{8 b^{3} \left (19 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{4389 c^{3} \sqrt{x}} + \frac{8 b^{2} x^{\frac{3}{2}} \left (19 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{7315 c^{2}} + \frac{4 b x^{\frac{7}{2}} \left (19 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{1045 c} + \frac{2 x^{\frac{3}{2}} \left (19 A c - 9 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{285 c} \]

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**(1/2),x)

[Out]

2*B*(b*x**2 + c*x**4)**(5/2)/(19*c*sqrt(x)) + 4*b**(15/4)*sqrt((b + c*x**2)/(sqr
t(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(19*A*c - 9*B*b)*sqrt(b*x**2 + c*x**
4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(4389*c**(13/4)*x*(b + c*x
**2)) - 8*b**3*(19*A*c - 9*B*b)*sqrt(b*x**2 + c*x**4)/(4389*c**3*sqrt(x)) + 8*b*
*2*x**(3/2)*(19*A*c - 9*B*b)*sqrt(b*x**2 + c*x**4)/(7315*c**2) + 4*b*x**(7/2)*(1
9*A*c - 9*B*b)*sqrt(b*x**2 + c*x**4)/(1045*c) + 2*x**(3/2)*(19*A*c - 9*B*b)*(b*x
**2 + c*x**4)**(3/2)/(285*c)

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Mathematica [C]  time = 0.689392, size = 198, normalized size = 0.7 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{-4 b^3 c \left (95 A+27 B x^2\right )+12 b^2 c^2 x^2 \left (19 A+7 B x^2\right )+7 b c^3 x^4 \left (323 A+231 B x^2\right )+77 c^4 x^6 \left (19 A+15 B x^2\right )+180 b^4 B}{\sqrt{x}}+\frac{20 i b^4 \sqrt{\frac{b}{c x^2}+1} (19 A c-9 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right )}{21945 c^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x^2*(b + c*x^2)]*((180*b^4*B + 12*b^2*c^2*x^2*(19*A + 7*B*x^2) + 77*c^4*
x^6*(19*A + 15*B*x^2) - 4*b^3*c*(95*A + 27*B*x^2) + 7*b*c^3*x^4*(323*A + 231*B*x
^2))/Sqrt[x] + ((20*I)*b^4*(-9*b*B + 19*A*c)*Sqrt[1 + b/(c*x^2)]*EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*
x^2))))/(21945*c^3)

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Maple [A]  time = 0.021, size = 331, normalized size = 1.2 \[{\frac{2}{21945\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 1155\,B{x}^{11}{c}^{6}+1463\,A{x}^{9}{c}^{6}+2772\,B{x}^{9}b{c}^{5}+3724\,A{x}^{7}b{c}^{5}+1701\,B{x}^{7}{b}^{2}{c}^{4}+190\,A\sqrt{-bc}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{b}^{4}c+2489\,A{x}^{5}{b}^{2}{c}^{4}-90\,B\sqrt{-bc}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{b}^{5}-24\,B{x}^{5}{b}^{3}{c}^{3}-152\,A{x}^{3}{b}^{3}{c}^{3}+72\,B{x}^{3}{b}^{4}{c}^{2}-380\,Ax{b}^{4}{c}^{2}+180\,Bx{b}^{5}c \right ){x}^{-{\frac{7}{2}}}} \]

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^(1/2),x)

[Out]

2/21945*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2*(1155*B*x^11*c^6+1463*A*x^9*c^6+
2772*B*x^9*b*c^5+3724*A*x^7*b*c^5+1701*B*x^7*b^2*c^4+190*A*(-b*c)^(1/2)*Elliptic
F(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*((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)*b^4*c+2489*A*x^5*b^2*c^4-90*B*(-b*c)^(1/2)*EllipticF(((c*x+(-b*c)^(1/2
))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*((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)*b^5-24*B*
x^5*b^3*c^3-152*A*x^3*b^3*c^3+72*B*x^3*b^4*c^2-380*A*x*b^4*c^2+180*B*x*b^5*c)/c^
4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*c*x^6 + (B*b + A*c)*x^4 + A*b*x^2)*sqrt(c*x^4 + b*x^2)*sqrt(x), x)

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

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**(1/2),x)

[Out]

Integral(sqrt(x)*(x**2*(b + c*x**2))**(3/2)*(A + B*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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