3.963 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=336 \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{3 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 a \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3} \]

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c^2*Sqrt[c + d*x^2])
- (2*a*(3*b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c^2*x) - (a*(a + b*x^2)
^(3/2)*Sqrt[c + d*x^2])/(3*c*x^3) - ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqrt[a
+ b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(3/2)*Sqr
t[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*(9*b*c - a*d)*S
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*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 = 0.825867, antiderivative size = 336, 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 \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{3 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 a \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c^2*Sqrt[c + d*x^2])
- (2*a*(3*b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c^2*x) - (a*(a + b*x^2)
^(3/2)*Sqrt[c + d*x^2])/(3*c*x^3) - ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqrt[a
+ b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(3/2)*Sqr
t[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*(9*b*c - a*d)*S
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*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 = 99.7162, size = 304, normalized size = 0.9 \[ - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{3 c x^{3}} + \frac{2 a \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - 3 b c\right )}{3 c^{2} x} - \frac{b \sqrt{a + b x^{2}} \left (a d - 9 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{x \sqrt{a + b x^{2}} \left (2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{3 c^{2} \sqrt{c + d x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 c^{\frac{3}{2}} \sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**4/(d*x**2+c)**(1/2),x)

[Out]

-a*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(3*c*x**3) + 2*a*sqrt(a + b*x**2)*sqrt(c
+ d*x**2)*(a*d - 3*b*c)/(3*c**2*x) - b*sqrt(a + b*x**2)*(a*d - 9*b*c)*elliptic_
f(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(3*sqrt(c)*sqrt(d)*sqrt(c*(a + b*x**2)
/(a*(c + d*x**2)))*sqrt(c + d*x**2)) - x*sqrt(a + b*x**2)*(2*a**2*d**2 - 7*a*b*c
*d - 3*b**2*c**2)/(3*c**2*sqrt(c + d*x**2)) + sqrt(a + b*x**2)*(2*a**2*d**2 - 7*
a*b*c*d - 3*b**2*c**2)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(3*c**
(3/2)*sqrt(d)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.814571, size = 261, normalized size = 0.78 \[ \frac{-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (2 a^2 d^2-7 a b c d-3 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+a d \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2-7 b c x^2\right )}{3 c^2 d x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]

[Out]

(a*Sqrt[b/a]*d*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 7*b*c*x^2 + 2*a*d*x^2) + I*b*c*
(-3*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]
*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-3*b^2*c^2 + 2*a*b*c*d
+ a^2*d^2)*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*d*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.026, size = 583, normalized size = 1.7 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ){c}^{2}{x}^{3}d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}{a}^{2}b{d}^{3}-7\,\sqrt{-{\frac{b}{a}}}{x}^{6}a{b}^{2}c{d}^{2}+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{a}^{2}bc{d}^{2}+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}a{b}^{2}{c}^{2}d-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{3}{c}^{3}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{a}^{2}bc{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}a{b}^{2}{c}^{2}d+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{3}{c}^{3}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{3}{d}^{3}-6\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}bc{d}^{2}-7\,\sqrt{-{\frac{b}{a}}}{x}^{4}a{b}^{2}{c}^{2}d+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{3}c{d}^{2}-8\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}b{c}^{2}d-\sqrt{-{\frac{b}{a}}}{a}^{3}{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/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^2*b*d^3-7*(-b/a)^(1/2)
*x^6*a*b^2*c*d^2+((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^2*b*c*d^2+2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ell
ipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*a*b^2*c^2*d-3*((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^3*c^3-2*((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
^2*b*c*d^2+7*((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^2*c^2*d+3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti
cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*b^3*c^3+2*(-b/a)^(1/2)*x^4*a^3*d^3-6*(-b/
a)^(1/2)*x^4*a^2*b*c*d^2-7*(-b/a)^(1/2)*x^4*a*b^2*c^2*d+(-b/a)^(1/2)*x^2*a^3*c*d
^2-8*(-b/a)^(1/2)*x^2*a^2*b*c^2*d-(-b/a)^(1/2)*a^3*c^2*d)/(b*d*x^4+a*d*x^2+b*c*x
^2+a*c)/c^2/x^3/(-b/a)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{x^{4} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**4/(d*x**2+c)**(1/2),x)

[Out]

Integral((a + b*x**2)**(5/2)/(x**4*sqrt(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4), x)