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

Optimal. Leaf size=330 \[ \frac{\sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{c}+7 a b-\frac{2 b^2 c}{d}\right )}{3 \sqrt{c+d x^2}}-\frac{b \sqrt{c} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{3 c d} \]

[Out]

((7*a*b - (2*b^2*c)/d + (3*a^2*d)/c)*x*Sqrt[a + b*x^2])/(3*Sqrt[c + d*x^2]) + (b
*(b*c + 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d) - (a*(a + b*x^2)^(3/2)
*Sqrt[c + d*x^2])/(c*x) + ((2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*E
llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*d^(3/2)*Sqrt[
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(b*c - 9*a*d)*Sqr
t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)
*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.781233, antiderivative size = 330, 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 \[ \frac{\sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{c}+7 a b-\frac{2 b^2 c}{d}\right )}{3 \sqrt{c+d x^2}}-\frac{b \sqrt{c} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{3 c d} \]

Antiderivative was successfully verified.

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

[Out]

((7*a*b - (2*b^2*c)/d + (3*a^2*d)/c)*x*Sqrt[a + b*x^2])/(3*Sqrt[c + d*x^2]) + (b
*(b*c + 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d) - (a*(a + b*x^2)^(3/2)
*Sqrt[c + d*x^2])/(c*x) + ((2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*E
llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*d^(3/2)*Sqrt[
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(b*c - 9*a*d)*Sqr
t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)
*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 = 99.0764, size = 308, normalized size = 0.93 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 c d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{c x} + \frac{b \sqrt{c} \sqrt{a + b x^{2}} \left (9 a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (3 a d + b c\right )}{3 c d} + \frac{b x \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right )}{3 c d^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*sqrt(b)*sqrt(c + d*x**2)*(3*a**2*d**2 + 7*a*b*c*d - 2*b**2*c**2)*ellipt
ic_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(3*c*d**2*sqrt(a*(c + d*x**2)/(c*(
a + b*x**2)))*sqrt(a + b*x**2)) - a*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(c*x) +
b*sqrt(c)*sqrt(a + b*x**2)*(9*a*d - b*c)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), 1
- b*c/(a*d))/(3*d**(3/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2))
+ b*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(3*a*d + b*c)/(3*c*d) + b*x*sqrt(c + d*
x**2)*(3*a**2*d**2 + 7*a*b*c*d - 2*b**2*c**2)/(3*c*d**2*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.788659, size = 254, normalized size = 0.77 \[ \frac{-2 i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d \left (-\sqrt{\frac{b}{a}}\right ) \left (a+b x^2\right ) \left (c+d x^2\right ) \left (3 a^2 d-b^2 c x^2\right )}{3 c d^2 x \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[b/a]*d*(a + b*x^2)*(3*a^2*d - b^2*c*x^2)*(c + d*x^2)) - I*b*c*(-2*b^2*c^
2 + 7*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I
*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*b*c*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2
)*x*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*d^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.027, size = 568, normalized size = 1.7 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ){d}^{2}cx}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{3}c{d}^{2}-3\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}b{d}^{3}+\sqrt{-{\frac{b}{a}}}{x}^{4}a{b}^{2}c{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{3}{c}^{2}d+6\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) x{a}^{2}bc{d}^{2}-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) xa{b}^{2}{c}^{2}d+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) x{b}^{3}{c}^{3}+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) x{a}^{2}bc{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) xa{b}^{2}{c}^{2}d-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) x{b}^{3}{c}^{3}-3\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{3}{d}^{3}-3\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}bc{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}a{b}^{2}{c}^{2}d-3\,\sqrt{-{\frac{b}{a}}}{a}^{3}c{d}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*((-b/a)^(1/2)*x^6*b^3*c*d^2-3*(-b/a)^(1/2)*x
^4*a^2*b*d^3+(-b/a)^(1/2)*x^4*a*b^2*c*d^2+(-b/a)^(1/2)*x^4*b^3*c^2*d+6*((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*a^2*b
*c*d^2-8*((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))*x*a*b^2*c^2*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*b^3*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))*x*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*a*b^2*c^2*d-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
*b^3*c^3-3*(-b/a)^(1/2)*x^2*a^3*d^3-3*(-b/a)^(1/2)*x^2*a^2*b*c*d^2+(-b/a)^(1/2)*
x^2*a*b^2*c^2*d-3*(-b/a)^(1/2)*a^3*c*d^2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/d^2/(-b/
a)^(1/2)/c/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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