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

Optimal. Leaf size=288 \[ -\frac{2 c^{7/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{5/2} \sqrt{c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{77 d^2 e}+\frac{4 c e \sqrt{e x} \sqrt{c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{231 d^3}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{55 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3} \]

[Out]

(4*c*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^3)
+ (2*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*d^2*e)
- (2*b*(3*b*c - 10*a*d)*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(55*d^2*e) + (2*b^2*(e*x
)^(9/2)*(c + d*x^2)^(3/2))/(15*d*e^3) - (2*c^(7/4)*(11*a^2*d^2 + b*c*(3*b*c - 10
*a*d))*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(13/4)*Sq
rt[c + d*x^2])

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Rubi [A]  time = 0.688208, antiderivative size = 288, 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 \[ -\frac{2 c^{7/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{13/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{5/2} \sqrt{c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{77 d^2 e}+\frac{4 c e \sqrt{e x} \sqrt{c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{231 d^3}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{55 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^3)
+ (2*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*d^2*e)
- (2*b*(3*b*c - 10*a*d)*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(55*d^2*e) + (2*b^2*(e*x
)^(9/2)*(c + d*x^2)^(3/2))/(15*d*e^3) - (2*c^(7/4)*(11*a^2*d^2 + b*c*(3*b*c - 10
*a*d))*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(13/4)*Sq
rt[c + d*x^2])

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Rubi in Sympy [A]  time = 57.6151, size = 277, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{9}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}{15 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (10 a d - 3 b c\right )}{55 d^{2} e} - \frac{2 c^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (11 a^{2} d^{2} - b c \left (10 a d - 3 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 d^{\frac{13}{4}} \sqrt{c + d x^{2}}} + \frac{4 c e \sqrt{e x} \sqrt{c + d x^{2}} \left (11 a^{2} d^{2} - b c \left (10 a d - 3 b c\right )\right )}{231 d^{3}} + \frac{2 \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (11 a^{2} d^{2} - b c \left (10 a d - 3 b c\right )\right )}{77 d^{2} e} \]

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)*(c + d*x**2)**(3/2)/(15*d*e**3) + 2*b*(e*x)**(5/2)*(c + d*x*
*2)**(3/2)*(10*a*d - 3*b*c)/(55*d**2*e) - 2*c**(7/4)*e**(3/2)*sqrt((c + d*x**2)/
(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(11*a**2*d**2 - b*c*(10*a*d - 3*
b*c))*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(231*d**(13
/4)*sqrt(c + d*x**2)) + 4*c*e*sqrt(e*x)*sqrt(c + d*x**2)*(11*a**2*d**2 - b*c*(10
*a*d - 3*b*c))/(231*d**3) + 2*(e*x)**(5/2)*sqrt(c + d*x**2)*(11*a**2*d**2 - b*c*
(10*a*d - 3*b*c))/(77*d**2*e)

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Mathematica [C]  time = 0.446055, size = 225, normalized size = 0.78 \[ \frac{(e x)^{3/2} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (55 a^2 d^2 \left (2 c+3 d x^2\right )+10 a b d \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )+b^2 \left (30 c^3-18 c^2 d x^2+14 c d^2 x^4+77 d^3 x^6\right )\right )}{5 d^3}-\frac{4 i c^2 x \sqrt{\frac{c}{d x^2}+1} \left (11 a^2 d^2-10 a b c d+3 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 x^{3/2} \sqrt{c+d x^2}} \]

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)*(55*a^2*d^2*(2*c + 3*d*x^2) + 10*a*b*d*(-10
*c^2 + 6*c*d*x^2 + 21*d^2*x^4) + b^2*(30*c^3 - 18*c^2*d*x^2 + 14*c*d^2*x^4 + 77*
d^3*x^6)))/(5*d^3) - ((4*I)*c^2*(3*b^2*c^2 - 10*a*b*c*d + 11*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 = 0.095, size = 448, normalized size = 1.6 \[ -{\frac{2\,e}{1155\,x{d}^{4}}\sqrt{ex} \left ( -77\,{x}^{9}{b}^{2}{d}^{5}-210\,{x}^{7}ab{d}^{5}-91\,{x}^{7}{b}^{2}c{d}^{4}+55\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{a}^{2}{c}^{2}{d}^{2}-50\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}ab{c}^{3}d+15\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{b}^{2}{c}^{4}-165\,{x}^{5}{a}^{2}{d}^{5}-270\,{x}^{5}abc{d}^{4}+4\,{x}^{5}{b}^{2}{c}^{2}{d}^{3}-275\,{x}^{3}{a}^{2}c{d}^{4}+40\,{x}^{3}ab{c}^{2}{d}^{3}-12\,{x}^{3}{b}^{2}{c}^{3}{d}^{2}-110\,x{a}^{2}{c}^{2}{d}^{3}+100\,xab{c}^{3}{d}^{2}-30\,x{b}^{2}{c}^{4}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-77*x^9*b^2*d^5-210*x^7*a*b*d^5-91*x^7*
b^2*c*d^4+55*((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))*(-c*d)^(1/2)*a^2*c^2*d^2-50*((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))*(-c
*d)^(1/2)*a*b*c^3*d+15*((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))*(-c*d)^(1/2)*b^2*c^4-165*x^5*a^2*d^5-27
0*x^5*a*b*c*d^4+4*x^5*b^2*c^2*d^3-275*x^3*a^2*c*d^4+40*x^3*a*b*c^2*d^3-12*x^3*b^
2*c^3*d^2-110*x*a^2*c^2*d^3+100*x*a*b*c^3*d^2-30*x*b^2*c^4*d)/d^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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