3.953 \(\int \frac{\left (a+b x^2\right )^{3/2}}{x^4 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=311 \[ -\frac{2 \sqrt{d} \sqrt{a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 c^2 \sqrt{c+d x^2}}+\frac{b \sqrt{a+b x^2} (3 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3} \]

[Out]

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

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Rubi [A]  time = 0.824706, antiderivative size = 311, 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 \[ -\frac{2 \sqrt{d} \sqrt{a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 c^2 \sqrt{c+d x^2}}+\frac{b \sqrt{a+b x^2} (3 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(2*b*c - a*d)*x*Sqrt[a + b*x^2])/(3*c^2*Sqrt[c + d*x^2]) - (a*Sqrt[a + b*x^
2]*Sqrt[c + d*x^2])/(3*c*x^3) - (2*(2*b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]
)/(3*c^2*x) - (2*Sqrt[d]*(2*b*c - a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]
*Sqrt[c + d*x^2]) + (b*(3*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x
)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*Sqrt[c]*Sqrt[d]*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 = 106.261, size = 279, normalized size = 0.9 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} \left (a d - 3 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} \left (a d - 2 b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{a \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{3 c x^{3}} - \frac{2 b x \sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{3 c^{2} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{3 c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)/x**4/(d*x**2+c)**(1/2),x)

[Out]

-sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*(a*d - 3*b*c)*elliptic_f(atan(sqrt(b)*x/sqrt(a
)), -a*d/(b*c) + 1)/(3*c**2*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**
2)) + 2*sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*(a*d - 2*b*c)*elliptic_e(atan(sqrt(b)*x
/sqrt(a)), -a*d/(b*c) + 1)/(3*c**2*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a
+ b*x**2)) - a*sqrt(a + b*x**2)*sqrt(c + d*x**2)/(3*c*x**3) - 2*b*x*sqrt(c + d*x
**2)*(a*d - 2*b*c)/(3*c**2*sqrt(a + b*x**2)) + 2*sqrt(a + b*x**2)*sqrt(c + d*x**
2)*(a*d - 2*b*c)/(3*c**2*x)

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Mathematica [C]  time = 0.605334, size = 227, normalized size = 0.73 \[ \frac{\sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2-4 b c x^2\right )-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 4*b*c*x^2 + 2*a*d*x^2) + (2*I)*b*c*
(-2*b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[S
qrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1
+ (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*x^
3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.027, size = 433, normalized size = 1.4 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ){c}^{2}{x}^{3}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}ab{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{2}cd+bd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}ac-\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}abcd+4\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}{d}^{2}-3\,\sqrt{-{\frac{b}{a}}}{x}^{4}abcd-4\,\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}cd-5\,\sqrt{-{\frac{b}{a}}}{x}^{2}ab{c}^{2}-\sqrt{-{\frac{b}{a}}}{a}^{2}{c}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/x^4/(d*x^2+c)^(1/2),x)

[Out]

1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(2*(-b/a)^(1/2)*x^6*a*b*d^2-4*(-b/a)^(1/2)*x
^6*b^2*c*d+b*d*((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))*x^3*a*c-((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))*x^3*b^2*c^2-2*((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))*x^3*a*b*c*d+4*((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))*x^3*b^2*c^2+2*(-b/a)^
(1/2)*x^4*a^2*d^2-3*(-b/a)^(1/2)*x^4*a*b*c*d-4*(-b/a)^(1/2)*x^4*b^2*c^2+(-b/a)^(
1/2)*x^2*a^2*c*d-5*(-b/a)^(1/2)*x^2*a*b*c^2-(-b/a)^(1/2)*a^2*c^2)/(b*d*x^4+a*d*x
^2+b*c*x^2+a*c)/c^2/x^3/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{d x^{2} + c} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)/x**4/(d*x**2+c)**(1/2),x)

[Out]

Integral((a + b*x**2)**(3/2)/(x**4*sqrt(c + d*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^4), x)