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

Optimal. Leaf size=403 \[ \frac{\sqrt{e x} \sqrt{c+d x^2} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right )}{2 c^2 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{(e x)^{3/2} (a d+3 b c) (b c-a d)}{2 c^2 d^2 e \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]

[Out]

((b*c - a*d)^2*(e*x)^(3/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(3*b*c
+ a*d)*(e*x)^(3/2))/(2*c^2*d^2*e*Sqrt[c + d*x^2]) + ((7*b^2*c^2 - 2*a*b*c*d - a^
2*d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(2*c^2*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - ((7*b^
2*c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqr
t[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
1/2])/(2*c^(7/4)*d^(11/4)*Sqrt[c + d*x^2]) + ((7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*
Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(4*c^(7/4)*d^(11/4)*Sqr
t[c + d*x^2])

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Rubi [A]  time = 0.821614, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt{e x} \sqrt{c+d x^2} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right )}{2 c^2 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt{c+d x^2}}-\frac{(e x)^{3/2} (a d+3 b c) (b c-a d)}{2 c^2 d^2 e \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(e*x)^(3/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(3*b*c
+ a*d)*(e*x)^(3/2))/(2*c^2*d^2*e*Sqrt[c + d*x^2]) + ((7*b^2*c^2 - 2*a*b*c*d - a^
2*d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(2*c^2*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - ((7*b^
2*c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqr
t[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
1/2])/(2*c^(7/4)*d^(11/4)*Sqrt[c + d*x^2]) + ((7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*
Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(4*c^(7/4)*d^(11/4)*Sqr
t[c + d*x^2])

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Rubi in Sympy [A]  time = 93.8246, size = 369, normalized size = 0.92 \[ \frac{\left (e x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{3 c d^{2} e \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{2 c^{2} d^{2} e \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (a^{2} d^{2} + 2 a b c d - 7 b^{2} c^{2}\right )}{2 c^{2} d^{\frac{5}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a^{2} d^{2} + 2 a b c d - 7 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 c^{\frac{7}{4}} d^{\frac{11}{4}} \sqrt{c + d x^{2}}} - \frac{\sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a^{2} d^{2} + 2 a b c d - 7 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 c^{\frac{7}{4}} d^{\frac{11}{4}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x)**(3/2)*(a*d - b*c)**2/(3*c*d**2*e*(c + d*x**2)**(3/2)) + (e*x)**(3/2)*(a*d
- b*c)*(a*d + 3*b*c)/(2*c**2*d**2*e*sqrt(c + d*x**2)) - sqrt(e*x)*sqrt(c + d*x*
*2)*(a**2*d**2 + 2*a*b*c*d - 7*b**2*c**2)/(2*c**2*d**(5/2)*(sqrt(c) + sqrt(d)*x)
) + sqrt(e)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a
**2*d**2 + 2*a*b*c*d - 7*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/
4)*sqrt(e))), 1/2)/(2*c**(7/4)*d**(11/4)*sqrt(c + d*x**2)) - sqrt(e)*sqrt((c + d
*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a**2*d**2 + 2*a*b*c*d -
7*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(4*c
**(7/4)*d**(11/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.33442, size = 281, normalized size = 0.7 \[ \frac{e \left (d x^2 \left (2 c (b c-a d)^2-3 \left (c+d x^2\right ) \left (-a^2 d^2-2 a b c d+3 b^2 c^2\right )\right )+\frac{3 \left (c+d x^2\right ) \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{6 c^2 d^3 \sqrt{e x} \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(d*x^2*(2*c*(b*c - a*d)^2 - 3*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*(c + d*x^2))
+ (3*(7*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*(c + d*x^2)*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(c
+ d*x^2) - Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt
[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] + Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2
)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/Sqrt[(I*Sqrt[c])
/Sqrt[d]]))/(6*c^2*d^3*Sqrt[e*x]*(c + d*x^2)^(3/2))

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Maple [B]  time = 0.033, size = 1176, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*((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^3+12*((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)*E
llipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^2*d^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*b^2*c^3*d-3*((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^2*c*d^3-6*((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^2+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)*EllipticF(((d*x+(-c
*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2*c^3*d+6*((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))*a^
2*c^2*d^2+12*((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))*a*b*c^3*d-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)*E
llipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4-3*((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))*a^2*c^2*d^2-6*((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))*a*b*c^3*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)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4
-6*x^4*a^2*d^4-12*x^4*a*b*c*d^3+18*x^4*b^2*c^2*d^2-10*x^2*a^2*c*d^3-4*x^2*a*b*c^
2*d^2+14*x^2*b^2*c^3*d)*(e*x)^(1/2)/d^3/c^2/x/(d*x^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(e*x)/((d^2*x^4 + 2*c*d*x^2 + c^2)*sqrt
(d*x^2 + c)), 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)**(1/2)/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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