3.794 \(\int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=252 \[ -\frac{4 a^{11/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 e \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{231 b^2}+\frac{4 a (e x)^{5/2} \sqrt{a+b x^2} (3 A b-a B)}{77 b e}+\frac{2 (e x)^{5/2} \left (a+b x^2\right )^{3/2} (3 A b-a B)}{33 b e}+\frac{2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e} \]

[Out]

(8*a^2*(3*A*b - a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(231*b^2) + (4*a*(3*A*b - a*B)
*(e*x)^(5/2)*Sqrt[a + b*x^2])/(77*b*e) + (2*(3*A*b - a*B)*(e*x)^(5/2)*(a + b*x^2
)^(3/2))/(33*b*e) + (2*B*(e*x)^(5/2)*(a + b*x^2)^(5/2))/(15*b*e) - (4*a^(11/4)*(
3*A*b - a*B)*e^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x
)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(231*b^(9/
4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.431151, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{4 a^{11/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 e \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{231 b^2}+\frac{4 a (e x)^{5/2} \sqrt{a+b x^2} (3 A b-a B)}{77 b e}+\frac{2 (e x)^{5/2} \left (a+b x^2\right )^{3/2} (3 A b-a B)}{33 b e}+\frac{2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^2*(3*A*b - a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(231*b^2) + (4*a*(3*A*b - a*B)
*(e*x)^(5/2)*Sqrt[a + b*x^2])/(77*b*e) + (2*(3*A*b - a*B)*(e*x)^(5/2)*(a + b*x^2
)^(3/2))/(33*b*e) + (2*B*(e*x)^(5/2)*(a + b*x^2)^(5/2))/(15*b*e) - (4*a^(11/4)*(
3*A*b - a*B)*e^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x
)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(231*b^(9/
4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 41.8079, size = 230, normalized size = 0.91 \[ \frac{2 B \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{5}{2}}}{15 b e} - \frac{4 a^{\frac{11}{4}} e^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} + \frac{8 a^{2} e \sqrt{e x} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{231 b^{2}} + \frac{4 a \left (e x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{77 b e} + \frac{2 \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 A b - B a\right )}{33 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)*(b*x**2+a)**(3/2)*(B*x**2+A),x)

[Out]

2*B*(e*x)**(5/2)*(a + b*x**2)**(5/2)/(15*b*e) - 4*a**(11/4)*e**(3/2)*sqrt((a + b
*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(3*A*b - B*a)*elliptic_f(
2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(231*b**(9/4)*sqrt(a + b*x**
2)) + 8*a**2*e*sqrt(e*x)*sqrt(a + b*x**2)*(3*A*b - B*a)/(231*b**2) + 4*a*(e*x)**
(5/2)*sqrt(a + b*x**2)*(3*A*b - B*a)/(77*b*e) + 2*(e*x)**(5/2)*(a + b*x**2)**(3/
2)*(3*A*b - B*a)/(33*b*e)

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Mathematica [C]  time = 0.585068, size = 178, normalized size = 0.71 \[ \frac{2 e \sqrt{e x} \left (-\left (a+b x^2\right ) \left (20 a^3 B-12 a^2 b \left (5 A+B x^2\right )-a b^2 x^2 \left (195 A+119 B x^2\right )-7 b^3 x^4 \left (15 A+11 B x^2\right )\right )+\frac{20 i a^3 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (a B-3 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{1155 b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[e*x]*(-((a + b*x^2)*(20*a^3*B - 12*a^2*b*(5*A + B*x^2) - 7*b^3*x^4*(15
*A + 11*B*x^2) - a*b^2*x^2*(195*A + 119*B*x^2))) + ((20*I)*a^3*(-3*A*b + a*B)*Sq
rt[1 + a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]]
, -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(1155*b^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.037, size = 300, normalized size = 1.2 \[ -{\frac{2\,e}{1155\,x{b}^{3}}\sqrt{ex} \left ( -77\,{b}^{5}B{x}^{9}-105\,A{x}^{7}{b}^{5}-196\,B{x}^{7}a{b}^{4}+30\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{3}b-300\,A{x}^{5}a{b}^{4}-10\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{4}-131\,B{x}^{5}{a}^{2}{b}^{3}-255\,A{x}^{3}{a}^{2}{b}^{3}+8\,B{x}^{3}{a}^{3}{b}^{2}-60\,Ax{a}^{3}{b}^{2}+20\,Bx{a}^{4}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x)

[Out]

-2/1155*e/x*(e*x)^(1/2)/(b*x^2+a)^(1/2)*(-77*b^5*B*x^9-105*A*x^7*b^5-196*B*x^7*a
*b^4+30*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(
-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b
)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a^3*b-300*A*x^5*a*b^4-10*B*((b*x+(-a*b)
^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x
*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(
1/2))*(-a*b)^(1/2)*a^4-131*B*x^5*a^2*b^3-255*A*x^3*a^2*b^3+8*B*x^3*a^3*b^2-60*A*
x*a^3*b^2+20*B*x*a^4*b)/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*b*e*x^5 + (B*a + A*b)*e*x^3 + A*a*e*x)*sqrt(b*x^2 + a)*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)**(3/2)*(B*x**2+A),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2), x)