3.205 \(\int \frac{\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=346 \[ -\frac{x \sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )}{3 c d^2 \sqrt{c+d x^2}}+\frac{\sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b \sqrt{c} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-3 a d)}{3 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]

[Out]

-((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c*d^2*Sqrt[c + d*x^
2]) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) + (b*(4*b*c - 3*a*
d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d^2) + ((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)])
/(3*Sqrt[c]*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*
b*Sqrt[c]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]],
1 - (b*c)/(a*d)])/(3*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 = 0.716793, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{x \sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )}{3 c d^2 \sqrt{c+d x^2}}+\frac{\sqrt{a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b \sqrt{c} \sqrt{a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (4 b c-3 a d)}{3 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2),x]

[Out]

-((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c*d^2*Sqrt[c + d*x^
2]) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) + (b*(4*b*c - 3*a*
d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d^2) + ((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)])
/(3*Sqrt[c]*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*
b*Sqrt[c]*(2*b*c - 3*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]],
1 - (b*c)/(a*d)])/(3*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 = 97.8097, size = 318, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \sqrt{b} \sqrt{c + d x^{2}} \left (3 a d - 2 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 c d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (3 a d - 4 b c\right )}{3 c d^{2}} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} - \frac{x \sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 13 a b c d + 8 b^{2} c^{2}\right )}{3 c d^{2} \sqrt{c + d x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 13 a b c d + 8 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 \sqrt{c} d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)

[Out]

2*a**(3/2)*sqrt(b)*sqrt(c + d*x**2)*(3*a*d - 2*b*c)*elliptic_f(atan(sqrt(b)*x/sq
rt(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)) - b*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(3*a*d - 4*b*c)/(3*c*d**2) + x
*(a + b*x**2)**(3/2)*(a*d - b*c)/(c*d*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)/(3*c*d**2*sqrt(c + d*x**2)) + 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))/(3*sqrt(c)*d**(5/2)*sqrt(c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(
c + d*x**2))

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Mathematica [C]  time = 0.822653, size = 256, normalized size = 0.74 \[ \frac{-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (9 a^2 d^2-17 a b c d+8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (3 a^2 d^2-6 a b c d+b^2 c \left (4 c+d x^2\right )\right )}{3 c d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2),x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(4*c + d*x^2)) + I*b*
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]*E
llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*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)])/(3*Sqrt[b/a]*c*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.036, size = 539, normalized size = 1.6 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ){d}^{3}c}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{3}c{d}^{2}+3\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}b{d}^{3}-5\,\sqrt{-{\frac{b}{a}}}{x}^{3}a{b}^{2}c{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{3}{c}^{2}d+9\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}bc{d}^{2}-17\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) a{b}^{2}{c}^{2}d+8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{3}{c}^{3}-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}bc{d}^{2}+13\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) a{b}^{2}{c}^{2}d-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{3}{c}^{3}+3\,x{a}^{3}{d}^{3}\sqrt{-{\frac{b}{a}}}-6\,\sqrt{-{\frac{b}{a}}}x{a}^{2}bc{d}^{2}+4\,\sqrt{-{\frac{b}{a}}}xa{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)

[Out]

1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*((-b/a)^(1/2)*x^5*b^3*c*d^2+3*(-b/a)^(1/2)*x
^3*a^2*b*d^3-5*(-b/a)^(1/2)*x^3*a*b^2*c*d^2+4*(-b/a)^(1/2)*x^3*b^3*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
*b*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^2*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^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))*a^2*b*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^2*c^2*d-8*((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))*b^3*c
^3+3*x*a^3*d^3*(-b/a)^(1/2)-6*(-b/a)^(1/2)*x*a^2*b*c*d^2+4*(-b/a)^(1/2)*x*a*b^2*
c^2*d)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/d^3/c/(-b/a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**(5/2)/(c + d*x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^(3/2), x)