3.856 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=434 \[ -\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*Sqrt[c + d*x^2]) - (2*a*(10*b*c - 7*a*d))/(5*c^2*e^3
*Sqrt[e*x]*Sqrt[c + d*x^2]) + ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(e*x)^(3/2))
/(5*c^3*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*Sqrt[e*x]*S
qrt[c + d*x^2])/(5*c^3*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^2*c^2 - 3*a*d*
(10*b*c - 7*a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2
]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(11/4)*d
^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(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])/(10*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d
*x^2])

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Rubi [A]  time = 1.01458, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*Sqrt[c + d*x^2]) - (2*a*(10*b*c - 7*a*d))/(5*c^2*e^3
*Sqrt[e*x]*Sqrt[c + d*x^2]) + ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(e*x)^(3/2))
/(5*c^3*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*Sqrt[e*x]*S
qrt[c + d*x^2])/(5*c^3*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^2*c^2 - 3*a*d*
(10*b*c - 7*a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2
]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(11/4)*d
^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(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])/(10*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d
*x^2])

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Rubi in Sympy [A]  time = 96.8345, size = 411, normalized size = 0.95 \[ - \frac{2 a^{2}}{5 c e \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}} + \frac{2 a \left (7 a d - 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x} \sqrt{c + d x^{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{5 c^{3} e^{5} \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{5 c^{3} \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{11}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (3 a d \left (7 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{11}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

-2*a**2/(5*c*e*(e*x)**(5/2)*sqrt(c + d*x**2)) + 2*a*(7*a*d - 10*b*c)/(5*c**2*e**
3*sqrt(e*x)*sqrt(c + d*x**2)) + (e*x)**(3/2)*(3*a*d*(7*a*d - 10*b*c) + 5*b**2*c*
*2)/(5*c**3*e**5*sqrt(c + d*x**2)) - sqrt(e*x)*sqrt(c + d*x**2)*(3*a*d*(7*a*d -
10*b*c) + 5*b**2*c**2)/(5*c**3*sqrt(d)*e**4*(sqrt(c) + sqrt(d)*x)) + sqrt((c + d
*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(3*a*d*(7*a*d - 10*b*c) +
5*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(5*
c**(11/4)*d**(3/4)*e**(7/2)*sqrt(c + d*x**2)) - sqrt((c + d*x**2)/(sqrt(c) + sqr
t(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(3*a*d*(7*a*d - 10*b*c) + 5*b**2*c**2)*ellipti
c_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(10*c**(11/4)*d**(3/4)*e
**(7/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.574978, size = 277, normalized size = 0.64 \[ \frac{i \left (\sqrt{d} \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \left (a^2 \left (-2 c^2+14 c d x^2+21 d^2 x^4\right )-10 a b c x^2 \left (2 c+3 d x^2\right )+5 b^2 c^2 x^4\right )+\sqrt{c} x^3 \sqrt{\frac{d x^2}{c}+1} \left (21 a^2 d^2-30 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-\sqrt{c} x^3 \sqrt{\frac{d x^2}{c}+1} \left (21 a^2 d^2-30 a b c d+5 b^2 c^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )\right )}{5 c^{7/2} e^2 (e x)^{3/2} \left (\frac{i \sqrt{d} x}{\sqrt{c}}\right )^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]

[Out]

((I/5)*(Sqrt[d]*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]*(5*b^2*c^2*x^4 - 10*a*b*c*x^2*(2*c +
3*d*x^2) + a^2*(-2*c^2 + 14*c*d*x^2 + 21*d^2*x^4)) - Sqrt[c]*(5*b^2*c^2 - 30*a*
b*c*d + 21*a^2*d^2)*x^3*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[d]*
x)/Sqrt[c]]], -1] + Sqrt[c]*(5*b^2*c^2 - 30*a*b*c*d + 21*a^2*d^2)*x^3*Sqrt[1 + (
d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1]))/(c^(7/2)*e^2*(
(I*Sqrt[d]*x)/Sqrt[c])^(3/2)*(e*x)^(3/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.036, size = 638, normalized size = 1.5 \[ -{\frac{1}{10\,d{x}^{2}{e}^{3}{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-60\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+30\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-42\,{x}^{4}{a}^{2}{d}^{3}+60\,{x}^{4}abc{d}^{2}-10\,{x}^{4}{b}^{2}{c}^{2}d-28\,{x}^{2}{a}^{2}c{d}^{2}+40\,{x}^{2}ab{c}^{2}d+4\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x)

[Out]

-1/10/x^2*(42*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(
-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^2-60*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^2*d
+10*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2
))^(1/2),1/2*2^(1/2))*x^2*b^2*c^3-21*((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)*Elliptic
F(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^2+30*((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))*x^2*a*b*c^2*d-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))*x^2*b^2*c^3-42*x^4*a^2*d^3+60*x^4*a*
b*c*d^2-10*x^4*b^2*c^2*d-28*x^2*a^2*c*d^2+40*x^2*a*b*c^2*d+4*a^2*c^2*d)/(d*x^2+c
)^(1/2)/d/e^3/(e*x)^(1/2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{{\left (d e^{3} x^{5} + c e^{3} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/((d*e^3*x^5 + c*e^3*x^3)*sqrt(d*x^2 + c)*sq
rt(e*x)), 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)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)