3.846 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=193 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-14 a b c d+21 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{7 c e (e x)^{7/2}}-\frac{2 a \sqrt{c+d x^2} (14 b c-5 a d)}{21 c^2 e^3 (e x)^{3/2}} \]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(7*c*e*(e*x)^(7/2)) - (2*a*(14*b*c - 5*a*d)*Sqrt[c + d*
x^2])/(21*c^2*e^3*(e*x)^(3/2)) + ((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*(Sqrt[c]
+ Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1
/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(21*c^(9/4)*d^(1/4)*e^(9/2)*Sqrt[c + d*
x^2])

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Rubi [A]  time = 0.432334, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-14 a b c d+21 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{7 c e (e x)^{7/2}}-\frac{2 a \sqrt{c+d x^2} (14 b c-5 a d)}{21 c^2 e^3 (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(7*c*e*(e*x)^(7/2)) - (2*a*(14*b*c - 5*a*d)*Sqrt[c + d*
x^2])/(21*c^2*e^3*(e*x)^(3/2)) + ((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*(Sqrt[c]
+ Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1
/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(21*c^(9/4)*d^(1/4)*e^(9/2)*Sqrt[c + d*
x^2])

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Rubi in Sympy [A]  time = 42.6298, size = 178, normalized size = 0.92 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{7 c e \left (e x\right )^{\frac{7}{2}}} + \frac{2 a \sqrt{c + d x^{2}} \left (5 a d - 14 b c\right )}{21 c^{2} e^{3} \left (e x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (5 a d - 14 b c\right ) + 21 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{9}{4}} \sqrt [4]{d} e^{\frac{9}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

-2*a**2*sqrt(c + d*x**2)/(7*c*e*(e*x)**(7/2)) + 2*a*sqrt(c + d*x**2)*(5*a*d - 14
*b*c)/(21*c**2*e**3*(e*x)**(3/2)) + sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*
(sqrt(c) + sqrt(d)*x)*(a*d*(5*a*d - 14*b*c) + 21*b**2*c**2)*elliptic_f(2*atan(d*
*(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(21*c**(9/4)*d**(1/4)*e**(9/2)*sqrt(c
+ d*x**2))

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Mathematica [C]  time = 0.303104, size = 159, normalized size = 0.82 \[ \frac{x^{9/2} \left (\frac{2 a \left (c+d x^2\right ) \left (-3 a c+5 a d x^2-14 b c x^2\right )}{c^2 x^{7/2}}+\frac{2 i x \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-14 a b c d+21 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 (e x)^{9/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

(x^(9/2)*((2*a*(c + d*x^2)*(-3*a*c - 14*b*c*x^2 + 5*a*d*x^2))/(c^2*x^(7/2)) + ((
2*I)*(21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]])))/
(21*(e*x)^(9/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.051, size = 370, normalized size = 1.9 \[{\frac{1}{21\,{x}^{3}d{c}^{2}{e}^{4}} \left ( 5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}{a}^{2}{d}^{2}-14\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}abcd+21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{3}{b}^{2}{c}^{2}+10\,{x}^{4}{a}^{2}{d}^{3}-28\,{x}^{4}abc{d}^{2}+4\,{x}^{2}{a}^{2}c{d}^{2}-28\,{x}^{2}ab{c}^{2}d-6\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x)

[Out]

1/21/(d*x^2+c)^(1/2)/x^3*(5*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d
*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(
-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^3*a^2*d^2-14*((d*x+
(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/
2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1
/2*2^(1/2))*(-c*d)^(1/2)*x^3*a*b*c*d+21*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*
2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*Ellip
ticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^3*b^2*c
^2+10*x^4*a^2*d^3-28*x^4*a*b*c*d^2+4*x^2*a^2*c*d^2-28*x^2*a*b*c^2*d-6*a^2*c^2*d)
/d/c^2/e^4/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/(sqrt(d*x^2 + c)*sqrt(e*x)*e^4*x^4), 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((b*x**2+a)**2/(e*x)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)), x)