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

Optimal. Leaf size=482 \[ \frac{4 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}-\frac{8 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{8 c^2 \sqrt{e x} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{1989 d^2 e}+\frac{4 c (e x)^{3/2} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^2 e}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3} \]

[Out]

(4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(3315*d
^2*e) + (8*c^2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])
/(3315*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d)
)*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2
)*(c + d*x^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e
^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e]*(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^(11/4)*Sqrt[c + d*x^2]) + (4*c^(9/4)*(2
21*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e]*(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)*Sqr
t[e])], 1/2])/(3315*d^(11/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.11401, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}-\frac{8 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{8 c^2 \sqrt{e x} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{1989 d^2 e}+\frac{4 c (e x)^{3/2} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^2 e}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(3315*d
^2*e) + (8*c^2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])
/(3315*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d)
)*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2
)*(c + d*x^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e
^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e]*(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^(11/4)*Sqrt[c + d*x^2]) + (4*c^(9/4)*(2
21*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*Sqrt[e]*(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)*Sqr
t[e])], 1/2])/(3315*d^(11/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 102.842, size = 464, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}}}{17 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (34 a d - 7 b c\right )}{221 d^{2} e} - \frac{8 c^{\frac{9}{4}} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (221 a^{2} d^{2} - 3 b c \left (34 a d - 7 b c\right )\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3315 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{4 c^{\frac{9}{4}} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (221 a^{2} d^{2} - 3 b c \left (34 a d - 7 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3315 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{8 c^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (221 a^{2} d^{2} - 3 b c \left (34 a d - 7 b c\right )\right )}{3315 d^{\frac{5}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{4 c \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (221 a^{2} d^{2} - 3 b c \left (34 a d - 7 b c\right )\right )}{3315 d^{2} e} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (221 a^{2} d^{2} - 3 b c \left (34 a d - 7 b c\right )\right )}{1989 d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)*(e*x)**(1/2),x)

[Out]

2*b**2*(e*x)**(7/2)*(c + d*x**2)**(5/2)/(17*d*e**3) + 2*b*(e*x)**(3/2)*(c + d*x*
*2)**(5/2)*(34*a*d - 7*b*c)/(221*d**2*e) - 8*c**(9/4)*sqrt(e)*sqrt((c + d*x**2)/
(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(221*a**2*d**2 - 3*b*c*(34*a*d -
7*b*c))*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(3315*d*
*(11/4)*sqrt(c + d*x**2)) + 4*c**(9/4)*sqrt(e)*sqrt((c + d*x**2)/(sqrt(c) + sqrt
(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(221*a**2*d**2 - 3*b*c*(34*a*d - 7*b*c))*ellipt
ic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(3315*d**(11/4)*sqrt(c
+ d*x**2)) + 8*c**2*sqrt(e*x)*sqrt(c + d*x**2)*(221*a**2*d**2 - 3*b*c*(34*a*d -
7*b*c))/(3315*d**(5/2)*(sqrt(c) + sqrt(d)*x)) + 4*c*(e*x)**(3/2)*sqrt(c + d*x**2
)*(221*a**2*d**2 - 3*b*c*(34*a*d - 7*b*c))/(3315*d**2*e) + 2*(e*x)**(3/2)*(c + d
*x**2)**(3/2)*(221*a**2*d**2 - 3*b*c*(34*a*d - 7*b*c))/(1989*d**2*e)

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Mathematica [C]  time = 1.46705, size = 316, normalized size = 0.66 \[ \frac{2 e \left (d x^2 \left (c+d x^2\right ) \left (221 a^2 d^2 \left (11 c+5 d x^2\right )+102 a b d \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+b^2 \left (-84 c^3+60 c^2 d x^2+855 c d^2 x^4+585 d^3 x^6\right )\right )+\frac{12 c^2 \left (221 a^2 d^2-102 a b c d+21 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{9945 d^3 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(d*x^2*(c + d*x^2)*(221*a^2*d^2*(11*c + 5*d*x^2) + 102*a*b*d*(4*c^2 + 25*c*
d*x^2 + 15*d^2*x^4) + b^2*(-84*c^3 + 60*c^2*d*x^2 + 855*c*d^2*x^4 + 585*d^3*x^6)
) + (12*c^2*(21*b^2*c^2 - 102*a*b*c*d + 221*a^2*d^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[Sq
rt[(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]]))/(9945*d^3*Sqrt[e*x]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.025, size = 699, normalized size = 1.5 \[{\frac{2}{9945\,{d}^{3}x}\sqrt{ex} \left ( 585\,{x}^{10}{b}^{2}{d}^{5}+1530\,{x}^{8}ab{d}^{5}+1440\,{x}^{8}{b}^{2}c{d}^{4}+1105\,{x}^{6}{a}^{2}{d}^{5}+4080\,{x}^{6}abc{d}^{4}+915\,{x}^{6}{b}^{2}{c}^{2}{d}^{3}+2652\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{3}{d}^{2}-1224\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{4}d+252\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{5}-1326\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{3}{d}^{2}+612\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{4}d-126\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{5}+3536\,{x}^{4}{a}^{2}c{d}^{4}+2958\,{x}^{4}ab{c}^{2}{d}^{3}-24\,{x}^{4}{b}^{2}{c}^{3}{d}^{2}+2431\,{x}^{2}{a}^{2}{c}^{2}{d}^{3}+408\,{x}^{2}ab{c}^{3}{d}^{2}-84\,{x}^{2}{b}^{2}{c}^{4}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x)

[Out]

2/9945/(d*x^2+c)^(1/2)*(e*x)^(1/2)/d^3*(585*x^10*b^2*d^5+1530*x^8*a*b*d^5+1440*x
^8*b^2*c*d^4+1105*x^6*a^2*d^5+4080*x^6*a*b*c*d^4+915*x^6*b^2*c^2*d^3+2652*((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^3*d^2-1224*((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^4*d+252*((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))*b
^2*c^5-1326*((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^3*d^2+612*((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^4*d-126*((
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^5+3536*x^4*a^2*c*d^4+2958*x^4*a*b*c^2*d^3-24*x^4*b^2*c^3*d
^2+2431*x^2*a^2*c^2*d^3+408*x^2*a*b*c^3*d^2-84*x^2*b^2*c^4*d)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*d*x^6 + (b^2*c + 2*a*b*d)*x^4 + a^2*c + (2*a*b*c + a^2*d)*x^2)*sqr
t(d*x^2 + c)*sqrt(e*x), x)

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Sympy [A]  time = 138.9, size = 304, normalized size = 0.63 \[ \frac{a^{2} c^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{a^{2} \sqrt{c} d \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{a b c^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{a b \sqrt{c} d \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{5} \Gamma \left (\frac{15}{4}\right )} + \frac{b^{2} c^{\frac{3}{2}} \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{4}\right )} + \frac{b^{2} \sqrt{c} d \left (e x\right )^{\frac{15}{2}} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{7} \Gamma \left (\frac{19}{4}\right )} \]

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)**(1/2),x)

[Out]

a**2*c**(3/2)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_pola
r(I*pi)/c)/(2*e*gamma(7/4)) + a**2*sqrt(c)*d*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2
, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**3*gamma(11/4)) + a*b*c**(3/2)*(
e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(e*
*3*gamma(11/4)) + a*b*sqrt(c)*d*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (1
5/4,), d*x**2*exp_polar(I*pi)/c)/(e**5*gamma(15/4)) + b**2*c**(3/2)*(e*x)**(11/2
)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**5*gam
ma(15/4)) + b**2*sqrt(c)*d*(e*x)**(15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,)
, d*x**2*exp_polar(I*pi)/c)/(2*e**7*gamma(19/4))

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GIAC/XCAS [A]  time = 0.432528, 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)*sqrt(e*x),x, algorithm="giac")

[Out]

Done