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

Optimal. Leaf size=240 \[ -\frac{c^{3/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 e \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right )}{231 d^3}-\frac{2 b (e x)^{5/2} \sqrt{c+d x^2} (9 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d e^3} \]

[Out]

(2*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^3)
- (2*b*(9*b*c - 22*a*d)*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*d^2*e) + (2*b^2*(e*x)^(
9/2)*Sqrt[c + d*x^2])/(11*d*e^3) - (c^(3/4)*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d)
)*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ellipt
icF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(13/4)*Sqrt[c
+ d*x^2])

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Rubi [A]  time = 0.552012, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{c^{3/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 e \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right )}{231 d^3}-\frac{2 b (e x)^{5/2} \sqrt{c+d x^2} (9 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^3)
- (2*b*(9*b*c - 22*a*d)*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*d^2*e) + (2*b^2*(e*x)^(
9/2)*Sqrt[c + d*x^2])/(11*d*e^3) - (c^(3/4)*(77*a^2*d^2 + 5*b*c*(9*b*c - 22*a*d)
)*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ellipt
icF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(13/4)*Sqrt[c
+ d*x^2])

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Rubi in Sympy [A]  time = 49.5247, size = 230, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{9}{2}} \sqrt{c + d x^{2}}}{11 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (22 a d - 9 b c\right )}{77 d^{2} e} - \frac{c^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (77 a^{2} d^{2} - 5 b c \left (22 a d - 9 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 d^{\frac{13}{4}} \sqrt{c + d x^{2}}} + \frac{2 e \sqrt{e x} \sqrt{c + d x^{2}} \left (77 a^{2} d^{2} - 5 b c \left (22 a d - 9 b c\right )\right )}{231 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(9/2)*sqrt(c + d*x**2)/(11*d*e**3) + 2*b*(e*x)**(5/2)*sqrt(c + d*x
**2)*(22*a*d - 9*b*c)/(77*d**2*e) - c**(3/4)*e**(3/2)*sqrt((c + d*x**2)/(sqrt(c)
+ sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(77*a**2*d**2 - 5*b*c*(22*a*d - 9*b*c))*
elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(231*d**(13/4)*sq
rt(c + d*x**2)) + 2*e*sqrt(e*x)*sqrt(c + d*x**2)*(77*a**2*d**2 - 5*b*c*(22*a*d -
9*b*c))/(231*d**3)

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Mathematica [C]  time = 0.345803, size = 190, normalized size = 0.79 \[ \frac{(e x)^{3/2} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (3 d x^2-5 c\right )+3 b^2 \left (15 c^2-9 c d x^2+7 d^2 x^4\right )\right )}{d^3}-\frac{2 i c x \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2-110 a b c d+45 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 x^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(77*a^2*d^2 + 22*a*b*d*(-5*c + 3*d*x^2) + 3
*b^2*(15*c^2 - 9*c*d*x^2 + 7*d^2*x^4)))/d^3 - ((2*I)*c*(45*b^2*c^2 - 110*a*b*c*d
+ 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d
]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(231*x^(3/2)*Sqrt[c + d*x^2]
)

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Maple [A]  time = 0.044, size = 405, normalized size = 1.7 \[ -{\frac{e}{231\,x{d}^{4}}\sqrt{ex} \left ( -42\,{x}^{7}{b}^{2}{d}^{4}+77\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-110\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+45\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-132\,{x}^{5}ab{d}^{4}+12\,{x}^{5}{b}^{2}c{d}^{3}-154\,{x}^{3}{a}^{2}{d}^{4}+88\,{x}^{3}abc{d}^{3}-36\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}-154\,x{a}^{2}c{d}^{3}+220\,xab{c}^{2}{d}^{2}-90\,x{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/231*e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-42*x^7*b^2*d^4+77*(-c*d)^(1/2)*((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*d^2-110*(-c*d)^(1/2)*((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))*a*b*c^2*d+45*(-c*d)^(1/2)
*((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^3-132*x^5*a*b*d^4+12*x^5*b^2*c*d^3-154*x^3*a^2*d^4+88*x
^3*a*b*c*d^3-36*x^3*b^2*c^2*d^2-154*x*a^2*c*d^3+220*x*a*b*c^2*d^2-90*x*b^2*c^3*d
)/d^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*e*x^5 + 2*a*b*e*x^3 + a^2*e*x)*sqrt(e*x)/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((e*x)**(3/2)*(b*x**2+a)**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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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