3.951 \(\int \frac{x^2 \left (a+b x^2\right )^{3/2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=335 \[ -\frac{\sqrt{c} \sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-\frac{3 a^2 d}{b}+13 a c-\frac{8 b c^2}{d}\right )}{15 d \sqrt{c+d x^2}}+\frac{2 c^{3/2} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{15 d^2}+\frac{b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 d} \]

[Out]

-((13*a*c - (8*b*c^2)/d - (3*a^2*d)/b)*x*Sqrt[a + b*x^2])/(15*d*Sqrt[c + d*x^2])
- (2*(2*b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (b*x^3*Sqrt[
a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (Sqrt[c]*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2
)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b
*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*c^(3/2)*(2*
b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a
*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.929073, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{c} \sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-\frac{3 a^2 d}{b}+13 a c-\frac{8 b c^2}{d}\right )}{15 d \sqrt{c+d x^2}}+\frac{2 c^{3/2} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-3 a d)}{15 d^2}+\frac{b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{5 d} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]

[Out]

-((13*a*c - (8*b*c^2)/d - (3*a^2*d)/b)*x*Sqrt[a + b*x^2])/(15*d*Sqrt[c + d*x^2])
- (2*(2*b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (b*x^3*Sqrt[
a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (Sqrt[c]*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2
)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b
*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*c^(3/2)*(2*
b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a
*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 106.902, size = 309, normalized size = 0.92 \[ \frac{b x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{5 d} - \frac{2 c^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (3 a d - 2 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{2 x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (3 a d - 2 b c\right )}{15 d^{2}} - \frac{\sqrt{c} \sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 13 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 b d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 13 a b c d + 8 b^{2} c^{2}\right )}{15 b d^{2} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

b*x**3*sqrt(a + b*x**2)*sqrt(c + d*x**2)/(5*d) - 2*c**(3/2)*sqrt(a + b*x**2)*(3*
a*d - 2*b*c)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(15*d**(5/2)*sqr
t(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) + 2*x*sqrt(a + b*x**2)*sqrt
(c + d*x**2)*(3*a*d - 2*b*c)/(15*d**2) - sqrt(c)*sqrt(a + b*x**2)*(3*a**2*d**2 -
13*a*b*c*d + 8*b**2*c**2)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(1
5*b*d**(5/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) + x*sqrt(a
+ b*x**2)*(3*a**2*d**2 - 13*a*b*c*d + 8*b**2*c**2)/(15*b*d**2*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.763775, size = 245, normalized size = 0.73 \[ \frac{i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (9 a^2 d^2-17 a b c d+8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a d-4 b c+3 b d x^2\right )}{15 d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + 6*a*d + 3*b*d*x^2) - I*c*(8*b^2
*c^2 - 13*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE
[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2)
*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)
/(b*c)])/(15*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.027, size = 544, normalized size = 1.6 \[ -{\frac{1}{15\,{d}^{3} \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) }\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( -3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{2}{d}^{3}-9\,\sqrt{-{\frac{b}{a}}}{x}^{5}ab{d}^{3}+\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{2}c{d}^{2}-6\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}{d}^{3}-5\,\sqrt{-{\frac{b}{a}}}{x}^{3}abc{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{2}{c}^{2}d+9\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}-17\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d+8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+13\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}-6\,\sqrt{-{\frac{b}{a}}}x{a}^{2}c{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}xab{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-3*(-b/a)^(1/2)*x^7*b^2*d^3-9*(-b/a)^(1/2
)*x^5*a*b*d^3+(-b/a)^(1/2)*x^5*b^2*c*d^2-6*(-b/a)^(1/2)*x^3*a^2*d^3-5*(-b/a)^(1/
2)*x^3*a*b*c*d^2+4*(-b/a)^(1/2)*x^3*b^2*c^2*d+9*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c
)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*c*d^2-17*((b*x^2+a)/a)^(1/
2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*d+8*((b
*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b
^2*c^3-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b
/c)^(1/2))*a^2*c*d^2+13*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/
a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*d-8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*El
lipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^3-6*(-b/a)^(1/2)*x*a^2*c*d^2+4*(-b
/a)^(1/2)*x*a*b*c^2*d)/d^3/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)*x^2/sqrt(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a*x^2)*sqrt(b*x^2 + a)/sqrt(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**2*(a + b*x**2)**(3/2)/sqrt(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)*x^2/sqrt(d*x^2 + c), x)