3.815 \(\int \frac{(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ \frac{5 e^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 \sqrt [4]{a} b^{13/4} \sqrt{a+b x^2}}-\frac{5 e^3 \sqrt{e x} (A b-3 a B)}{6 b^3 \sqrt{a+b x^2}}-\frac{e (e x)^{5/2} (A b-3 a B)}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac{2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}} \]

[Out]

-((A*b - 3*a*B)*e*(e*x)^(5/2))/(3*b^2*(a + b*x^2)^(3/2)) + (2*B*(e*x)^(9/2))/(3*
b*e*(a + b*x^2)^(3/2)) - (5*(A*b - 3*a*B)*e^3*Sqrt[e*x])/(6*b^3*Sqrt[a + b*x^2])
+ (5*(A*b - 3*a*B)*e^(7/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sq
rt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(12
*a^(1/4)*b^(13/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.358899, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{5 e^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 \sqrt [4]{a} b^{13/4} \sqrt{a+b x^2}}-\frac{5 e^3 \sqrt{e x} (A b-3 a B)}{6 b^3 \sqrt{a+b x^2}}-\frac{e (e x)^{5/2} (A b-3 a B)}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac{2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

-((A*b - 3*a*B)*e*(e*x)^(5/2))/(3*b^2*(a + b*x^2)^(3/2)) + (2*B*(e*x)^(9/2))/(3*
b*e*(a + b*x^2)^(3/2)) - (5*(A*b - 3*a*B)*e^3*Sqrt[e*x])/(6*b^3*Sqrt[a + b*x^2])
+ (5*(A*b - 3*a*B)*e^(7/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sq
rt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(12
*a^(1/4)*b^(13/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 38.1814, size = 194, normalized size = 0.93 \[ \frac{2 B \left (e x\right )^{\frac{9}{2}}}{3 b e \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{e \left (e x\right )^{\frac{5}{2}} \left (A b - 3 B a\right )}{3 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{5 e^{3} \sqrt{e x} \left (A b - 3 B a\right )}{6 b^{3} \sqrt{a + b x^{2}}} + \frac{5 e^{\frac{7}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt [4]{a} b^{\frac{13}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

2*B*(e*x)**(9/2)/(3*b*e*(a + b*x**2)**(3/2)) - e*(e*x)**(5/2)*(A*b - 3*B*a)/(3*b
**2*(a + b*x**2)**(3/2)) - 5*e**3*sqrt(e*x)*(A*b - 3*B*a)/(6*b**3*sqrt(a + b*x**
2)) + 5*e**(7/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*
x)*(A*b - 3*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/
(12*a**(1/4)*b**(13/4)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.392782, size = 163, normalized size = 0.78 \[ \frac{(e x)^{7/2} \left (\frac{\sqrt{x} \left (15 a^2 B+a \left (21 b B x^2-5 A b\right )+b^2 x^2 \left (4 B x^2-7 A\right )\right )}{b^3 \left (a+b x^2\right )}+\frac{5 i x \sqrt{\frac{a}{b x^2}+1} (A b-3 a B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{b^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{6 x^{7/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

((e*x)^(7/2)*((Sqrt[x]*(15*a^2*B + b^2*x^2*(-7*A + 4*B*x^2) + a*(-5*A*b + 21*b*B
*x^2)))/(b^3*(a + b*x^2)) + ((5*I)*(A*b - 3*a*B)*Sqrt[1 + a/(b*x^2)]*x*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[a])/Sqrt[b]]*b
^3)))/(6*x^(7/2)*Sqrt[a + b*x^2])

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Maple [B]  time = 0.056, size = 439, normalized size = 2.1 \[{\frac{{e}^{3}}{12\,x{b}^{4}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{2}{b}^{2}-15\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{2}ab+5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}ab-15\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{2}+8\,B{x}^{5}{b}^{3}-14\,A{x}^{3}{b}^{3}+42\,B{x}^{3}a{b}^{2}-10\,Axa{b}^{2}+30\,Bx{a}^{2}b \right ) \sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

1/12*(5*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)*x^2*b^2-15*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)*x^2*a*b+5*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*b-15*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^2+8*B*x^5*b^3-14*A*x^3*b^3+42*B*x^3*a*b^2-10*A*x*a*b^2+30*B*x*a^2*b)*e
^3/x*(e*x)^(1/2)/b^4/(b*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{5} + A e^{3} x^{3}\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^5 + A*e^3*x^3)*sqrt(e*x)/((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x
^2 + a)), 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)**(7/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(5/2), x)