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

Optimal. Leaf size=530 \[ -\frac{4 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}-\frac{8 c^3 e^2 \sqrt{e x} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{8 c^2 e (e x)^{3/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac{2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac{4 c (e x)^{7/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac{2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]

[Out]

(8*c^2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(9945
*d^3) + (4*c*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e*x)^(7/2)*Sqrt[c + d*x^2])/(
1989*d^2*e) - (8*c^3*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^2*Sqrt[e*x]*Sqrt[c +
d*x^2])/(3315*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(51*a^2*d^2 + b*c*(11*b*c - 4
2*a*d))*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(663*d^2*e) - (2*b*(11*b*c - 42*a*d)*(e*x
)^(7/2)*(c + d*x^2)^(5/2))/(357*d^2*e) + (2*b^2*(e*x)^(11/2)*(c + d*x^2)^(5/2))/
(21*d*e^3) + (8*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(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])/(3315*d^(15/4)*Sqrt[c + d*x^2]) - (4*c^(1
3/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(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])/(3315*d^(15/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.28858, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{4 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}-\frac{8 c^3 e^2 \sqrt{e x} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{8 c^2 e (e x)^{3/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac{2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac{4 c (e x)^{7/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac{2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(8*c^2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(9945
*d^3) + (4*c*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e*x)^(7/2)*Sqrt[c + d*x^2])/(
1989*d^2*e) - (8*c^3*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^2*Sqrt[e*x]*Sqrt[c +
d*x^2])/(3315*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(51*a^2*d^2 + b*c*(11*b*c - 4
2*a*d))*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(663*d^2*e) - (2*b*(11*b*c - 42*a*d)*(e*x
)^(7/2)*(c + d*x^2)^(5/2))/(357*d^2*e) + (2*b^2*(e*x)^(11/2)*(c + d*x^2)^(5/2))/
(21*d*e^3) + (8*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(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])/(3315*d^(15/4)*Sqrt[c + d*x^2]) - (4*c^(1
3/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*(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])/(3315*d^(15/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 119.482, size = 510, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{11}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}}}{21 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{7}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (42 a d - 11 b c\right )}{357 d^{2} e} + \frac{8 c^{\frac{13}{4}} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3315 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{4 c^{\frac{13}{4}} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3315 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{8 c^{3} e^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right )}{3315 d^{\frac{7}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{8 c^{2} e \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right )}{9945 d^{3}} + \frac{4 c \left (e x\right )^{\frac{7}{2}} \sqrt{c + d x^{2}} \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right )}{1989 d^{2} e} + \frac{2 \left (e x\right )^{\frac{7}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (51 a^{2} d^{2} - b c \left (42 a d - 11 b c\right )\right )}{663 d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(11/2)*(c + d*x**2)**(5/2)/(21*d*e**3) + 2*b*(e*x)**(7/2)*(c + d*x
**2)**(5/2)*(42*a*d - 11*b*c)/(357*d**2*e) + 8*c**(13/4)*e**(5/2)*sqrt((c + d*x*
*2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(51*a**2*d**2 - b*c*(42*a*d
- 11*b*c))*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(3315*
d**(15/4)*sqrt(c + d*x**2)) - 4*c**(13/4)*e**(5/2)*sqrt((c + d*x**2)/(sqrt(c) +
sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(51*a**2*d**2 - b*c*(42*a*d - 11*b*c))*elli
ptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(3315*d**(15/4)*sqrt(
c + d*x**2)) - 8*c**3*e**2*sqrt(e*x)*sqrt(c + d*x**2)*(51*a**2*d**2 - b*c*(42*a*
d - 11*b*c))/(3315*d**(7/2)*(sqrt(c) + sqrt(d)*x)) + 8*c**2*e*(e*x)**(3/2)*sqrt(
c + d*x**2)*(51*a**2*d**2 - b*c*(42*a*d - 11*b*c))/(9945*d**3) + 4*c*(e*x)**(7/2
)*sqrt(c + d*x**2)*(51*a**2*d**2 - b*c*(42*a*d - 11*b*c))/(1989*d**2*e) + 2*(e*x
)**(7/2)*(c + d*x**2)**(3/2)*(51*a**2*d**2 - b*c*(42*a*d - 11*b*c))/(663*d**2*e)

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Mathematica [C]  time = 2.04367, size = 304, normalized size = 0.57 \[ \frac{2 (e x)^{5/2} \left (d \sqrt{x} \left (c+d x^2\right ) \left (357 a^2 d^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+42 a b d \left (-28 c^3+20 c^2 d x^2+285 c d^2 x^4+195 d^3 x^6\right )+b^2 \left (308 c^4-220 c^3 d x^2+180 c^2 d^2 x^4+4485 c d^3 x^6+3315 d^4 x^8\right )\right )+84 c^3 \left (51 a^2 d^2-42 a b c d+11 b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )\right )}{69615 d^4 x^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*x)^(5/2)*(d*Sqrt[x]*(c + d*x^2)*(357*a^2*d^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*
x^4) + 42*a*b*d*(-28*c^3 + 20*c^2*d*x^2 + 285*c*d^2*x^4 + 195*d^3*x^6) + b^2*(30
8*c^4 - 220*c^3*d*x^2 + 180*c^2*d^2*x^4 + 4485*c*d^3*x^6 + 3315*d^4*x^8)) + 84*c
^3*(11*b^2*c^2 - 42*a*b*c*d + 51*a^2*d^2)*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1
+ c/(d*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - Elli
pticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^
(3/2))))/(69615*d^4*x^(3/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.054, size = 743, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/69615/x*e^2*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^4*(-3315*x^12*b^2*d^6-8190*x^10*a*b
*d^6-7800*x^10*b^2*c*d^5-5355*x^8*a^2*d^6-20160*x^8*a*b*c*d^5-4665*x^8*b^2*c^2*d
^4-14280*x^6*a^2*c*d^5-12810*x^6*a*b*c^2*d^4+40*x^6*b^2*c^3*d^3+4284*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))*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))
^(1/2)*a^2*c^4*d^2-3528*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)
)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*a*b*c^5*d+924*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))*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*b^2*c^
6-2142*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))*((d*x+(-c*d)^(1
/2))/(-c*d)^(1/2))^(1/2)*a^2*c^4*d^2+1764*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))*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*a*b*c^5*d-462*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))*((d*x+(-c*d)^(1/2))/(-c*d)^(
1/2))^(1/2)*b^2*c^6-10353*x^4*a^2*c^2*d^4+336*x^4*a*b*c^3*d^3-88*x^4*b^2*c^4*d^2
-1428*x^2*a^2*c^3*d^3+1176*x^2*a*b*c^4*d^2-308*x^2*b^2*c^5*d)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} d e^{2} x^{8} +{\left (b^{2} c + 2 \, a b d\right )} e^{2} x^{6} + a^{2} c e^{2} x^{2} +{\left (2 \, a b c + a^{2} d\right )} e^{2} x^{4}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*d*e^2*x^8 + (b^2*c + 2*a*b*d)*e^2*x^6 + a^2*c*e^2*x^2 + (2*a*b*c +
a^2*d)*e^2*x^4)*sqrt(d*x^2 + c)*sqrt(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((e*x)**(5/2)*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.450555, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^(5/2),x, algorithm="giac")

[Out]

Done