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

Optimal. Leaf size=340 \[ -\frac{4 c^{11/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (57 a^2 d^2+b c (9 b c-38 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4389 d^{13/4} \sqrt{c+d x^2}}+\frac{8 c^2 e \sqrt{e x} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{4389 d^3}+\frac{2 (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{627 d^2 e}+\frac{4 c (e x)^{5/2} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{1463 d^2 e}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{5/2} (9 b c-38 a d)}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3} \]

[Out]

(8*c^2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(4389*d^
3) + (4*c*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*Sqrt[c + d*x^2])/(1463
*d^2*e) + (2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*(c + d*x^2)^(3/2))/
(627*d^2*e) - (2*b*(9*b*c - 38*a*d)*(e*x)^(5/2)*(c + d*x^2)^(5/2))/(285*d^2*e) +
(2*b^2*(e*x)^(9/2)*(c + d*x^2)^(5/2))/(19*d*e^3) - (4*c^(11/4)*(57*a^2*d^2 + b*
c*(9*b*c - 38*a*d))*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sq
rt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(43
89*d^(13/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.779461, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{4 c^{11/4} e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (57 a^2 d^2+b c (9 b c-38 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4389 d^{13/4} \sqrt{c+d x^2}}+\frac{8 c^2 e \sqrt{e x} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{4389 d^3}+\frac{2 (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{627 d^2 e}+\frac{4 c (e x)^{5/2} \sqrt{c+d x^2} \left (57 a^2 d^2+b c (9 b c-38 a d)\right )}{1463 d^2 e}-\frac{2 b (e x)^{5/2} \left (c+d x^2\right )^{5/2} (9 b c-38 a d)}{285 d^2 e}+\frac{2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{5/2}}{19 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(8*c^2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(4389*d^
3) + (4*c*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*Sqrt[c + d*x^2])/(1463
*d^2*e) + (2*(57*a^2*d^2 + b*c*(9*b*c - 38*a*d))*(e*x)^(5/2)*(c + d*x^2)^(3/2))/
(627*d^2*e) - (2*b*(9*b*c - 38*a*d)*(e*x)^(5/2)*(c + d*x^2)^(5/2))/(285*d^2*e) +
(2*b^2*(e*x)^(9/2)*(c + d*x^2)^(5/2))/(19*d*e^3) - (4*c^(11/4)*(57*a^2*d^2 + b*
c*(9*b*c - 38*a*d))*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sq
rt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(43
89*d^(13/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 67.0829, size = 328, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{9}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}}}{19 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (38 a d - 9 b c\right )}{285 d^{2} e} - \frac{4 c^{\frac{11}{4}} e^{\frac{3}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (57 a^{2} d^{2} - b c \left (38 a d - 9 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4389 d^{\frac{13}{4}} \sqrt{c + d x^{2}}} + \frac{8 c^{2} e \sqrt{e x} \sqrt{c + d x^{2}} \left (57 a^{2} d^{2} - b c \left (38 a d - 9 b c\right )\right )}{4389 d^{3}} + \frac{4 c \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (57 a^{2} d^{2} - b c \left (38 a d - 9 b c\right )\right )}{1463 d^{2} e} + \frac{2 \left (e x\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (57 a^{2} d^{2} - b c \left (38 a d - 9 b c\right )\right )}{627 d^{2} e} \]

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)**(3/2),x)

[Out]

2*b**2*(e*x)**(9/2)*(c + d*x**2)**(5/2)/(19*d*e**3) + 2*b*(e*x)**(5/2)*(c + d*x*
*2)**(5/2)*(38*a*d - 9*b*c)/(285*d**2*e) - 4*c**(11/4)*e**(3/2)*sqrt((c + d*x**2
)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(57*a**2*d**2 - b*c*(38*a*d -
9*b*c))*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(4389*d**
(13/4)*sqrt(c + d*x**2)) + 8*c**2*e*sqrt(e*x)*sqrt(c + d*x**2)*(57*a**2*d**2 - b
*c*(38*a*d - 9*b*c))/(4389*d**3) + 4*c*(e*x)**(5/2)*sqrt(c + d*x**2)*(57*a**2*d*
*2 - b*c*(38*a*d - 9*b*c))/(1463*d**2*e) + 2*(e*x)**(5/2)*(c + d*x**2)**(3/2)*(5
7*a**2*d**2 - b*c*(38*a*d - 9*b*c))/(627*d**2*e)

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Mathematica [C]  time = 0.442885, size = 259, normalized size = 0.76 \[ \frac{(e x)^{3/2} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (285 a^2 d^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )+38 a b d \left (-20 c^3+12 c^2 d x^2+119 c d^2 x^4+77 d^3 x^6\right )+3 b^2 \left (60 c^4-36 c^3 d x^2+28 c^2 d^2 x^4+539 c d^3 x^6+385 d^4 x^8\right )\right )}{5 d^3}-\frac{8 i c^3 x \sqrt{\frac{c}{d x^2}+1} \left (57 a^2 d^2-38 a b c d+9 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{4389 x^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(285*a^2*d^2*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^
4) + 38*a*b*d*(-20*c^3 + 12*c^2*d*x^2 + 119*c*d^2*x^4 + 77*d^3*x^6) + 3*b^2*(60*
c^4 - 36*c^3*d*x^2 + 28*c^2*d^2*x^4 + 539*c*d^3*x^6 + 385*d^4*x^8)))/(5*d^3) - (
(8*I)*c^3*(9*b^2*c^2 - 38*a*b*c*d + 57*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)))/(4389*x^(3/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.041, size = 489, normalized size = 1.4 \[ -{\frac{2\,e}{21945\,x{d}^{4}}\sqrt{ex} \left ( -1155\,{x}^{11}{b}^{2}{d}^{6}-2926\,{x}^{9}ab{d}^{6}-2772\,{x}^{9}{b}^{2}c{d}^{5}-1995\,{x}^{7}{a}^{2}{d}^{6}-7448\,{x}^{7}abc{d}^{5}-1701\,{x}^{7}{b}^{2}{c}^{2}{d}^{4}+570\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{a}^{2}{c}^{3}{d}^{2}-380\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}ab{c}^{4}d+90\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{b}^{2}{c}^{5}-5700\,{x}^{5}{a}^{2}c{d}^{5}-4978\,{x}^{5}ab{c}^{2}{d}^{4}+24\,{x}^{5}{b}^{2}{c}^{3}{d}^{3}-4845\,{x}^{3}{a}^{2}{c}^{2}{d}^{4}+304\,{x}^{3}ab{c}^{3}{d}^{3}-72\,{x}^{3}{b}^{2}{c}^{4}{d}^{2}-1140\,x{a}^{2}{c}^{3}{d}^{3}+760\,xab{c}^{4}{d}^{2}-180\,x{b}^{2}{c}^{5}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

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)^(3/2),x)

[Out]

-2/21945*e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-1155*x^11*b^2*d^6-2926*x^9*a*b*d^6-27
72*x^9*b^2*c*d^5-1995*x^7*a^2*d^6-7448*x^7*a*b*c*d^5-1701*x^7*b^2*c^2*d^4+570*((
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^3*d^2-380*((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^4*d+90*((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^5-5700*x^5*a^2*c*d^5-4978*x^5*a*b*
c^2*d^4+24*x^5*b^2*c^3*d^3-4845*x^3*a^2*c^2*d^4+304*x^3*a*b*c^3*d^3-72*x^3*b^2*c
^4*d^2-1140*x*a^2*c^3*d^3+760*x*a*b*c^4*d^2-180*x*b^2*c^5*d)/d^4

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

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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} d e x^{7} +{\left (b^{2} c + 2 \, a b d\right )} e x^{5} + a^{2} c e x +{\left (2 \, a b c + a^{2} d\right )} e x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, x\right ) \]

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)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*d*e*x^7 + (b^2*c + 2*a*b*d)*e*x^5 + a^2*c*e*x + (2*a*b*c + a^2*d)*
e*x^3)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.339373, 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)^(3/2),x, algorithm="giac")

[Out]

Done