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

Optimal. Leaf size=229 \[ \frac{2 \sqrt{c+d x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 b \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.37155, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2 \sqrt{c+d x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 b \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{c+d x^2} (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*x*Sqrt[c + d*x^2])/(3*a*b*(a + b*x^2)^(3/2)) + (2*(b*c + a*d)*Sqrt[
c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*a^(3/2)*b
^(3/2)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (c^(3/2)*Sqrt[d]
*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2
*b*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 = 46.2652, size = 192, normalized size = 0.84 \[ - \frac{x \sqrt{c + d x^{2}} \left (a d - b c\right )}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{d \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 \sqrt{a} b^{\frac{3}{2}} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{c + d x^{2}} \left (a d + b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 a^{\frac{3}{2}} b^{\frac{3}{2}} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.800394, size = 232, normalized size = 1.01 \[ \frac{x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (a^2 d+a b \left (3 c+2 d x^2\right )+2 b^2 c x^2\right )-i c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 a^3 \left (\frac{b}{a}\right )^{3/2} \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 607, normalized size = 2.7 \[{\frac{1}{3\,{a}^{2}b} \left ( 2\,{x}^{5}ab{d}^{2}\sqrt{-{\frac{b}{a}}}+2\,{x}^{5}{b}^{2}cd\sqrt{-{\frac{b}{a}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}abcd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}abcd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{x}^{3}{a}^{2}{d}^{2}\sqrt{-{\frac{b}{a}}}+5\,{x}^{3}abcd\sqrt{-{\frac{b}{a}}}+2\,{x}^{3}{b}^{2}{c}^{2}\sqrt{-{\frac{b}{a}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}cd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}cd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+x{a}^{2}cd\sqrt{-{\frac{b}{a}}}+3\,xab{c}^{2}\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{-{\frac{b}{a}}}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*x^5*a*b*d^2*(-b/a)^(1/2)+2*x^5*b^2*c*d*(-b/a)^(1/2)+EllipticF(x*(-b/a)^(1
/2),(a*d/b/c)^(1/2))*x^2*a*b*c*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+2*Ellip
ticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*b^2*c^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/
c)^(1/2)-2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a*b*c*d*((b*x^2+a)/a)^(
1/2)*((d*x^2+c)/c)^(1/2)-2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*b^2*c^2
*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+x^3*a^2*d^2*(-b/a)^(1/2)+5*x^3*a*b*c*d*
(-b/a)^(1/2)+2*x^3*b^2*c^2*(-b/a)^(1/2)+EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)
)*a^2*c*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+2*EllipticF(x*(-b/a)^(1/2),(a*
d/b/c)^(1/2))*a*b*c^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-2*EllipticE(x*(-b/
a)^(1/2),(a*d/b/c)^(1/2))*a^2*c*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-2*Elli
pticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^
(1/2)+x*a^2*c*d*(-b/a)^(1/2)+3*x*a*b*c^2*(-b/a)^(1/2))/(d*x^2+c)^(1/2)/a^2/(-b/a
)^(1/2)/(b*x^2+a)^(3/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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