3.857 \(\int \frac{(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{5 e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt{c+d x^2}}-\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac{e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}} \]

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((39*b^2*c^2 - 42*a*
b*c*d + 7*a^2*d^2)*e*(e*x)^(5/2))/(14*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(9/2
))/(7*d^2*e*Sqrt[c + d*x^2]) - (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^3*Sqrt
[e*x]*Sqrt[c + d*x^2])/(42*c*d^4) + (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^(
7/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2
*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(84*c^(1/4)*d^(17/4)*Sqrt[
c + d*x^2])

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Rubi [A]  time = 0.610062, antiderivative size = 302, 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 \[ \frac{5 e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt{c+d x^2}}-\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac{e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((39*b^2*c^2 - 42*a*
b*c*d + 7*a^2*d^2)*e*(e*x)^(5/2))/(14*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(9/2
))/(7*d^2*e*Sqrt[c + d*x^2]) - (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^3*Sqrt
[e*x]*Sqrt[c + d*x^2])/(42*c*d^4) + (5*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e^(
7/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2
*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(84*c^(1/4)*d^(17/4)*Sqrt[
c + d*x^2])

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Rubi in Sympy [A]  time = 71.672, size = 286, normalized size = 0.95 \[ \frac{2 b^{2} \left (e x\right )^{\frac{9}{2}}}{7 d^{2} e \sqrt{c + d x^{2}}} + \frac{\left (e x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}}{3 c d^{2} e \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{e \left (e x\right )^{\frac{5}{2}} \left (7 a^{2} d^{2} - 42 a b c d + 39 b^{2} c^{2}\right )}{14 c d^{3} \sqrt{c + d x^{2}}} - \frac{5 e^{3} \sqrt{e x} \sqrt{c + d x^{2}} \left (7 a^{2} d^{2} - 42 a b c d + 39 b^{2} c^{2}\right )}{42 c d^{4}} + \frac{5 e^{\frac{7}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (7 a^{2} d^{2} - 42 a b c d + 39 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{84 \sqrt [4]{c} d^{\frac{17}{4}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(9/2)/(7*d**2*e*sqrt(c + d*x**2)) + (e*x)**(9/2)*(a*d - b*c)**2/(3
*c*d**2*e*(c + d*x**2)**(3/2)) + e*(e*x)**(5/2)*(7*a**2*d**2 - 42*a*b*c*d + 39*b
**2*c**2)/(14*c*d**3*sqrt(c + d*x**2)) - 5*e**3*sqrt(e*x)*sqrt(c + d*x**2)*(7*a*
*2*d**2 - 42*a*b*c*d + 39*b**2*c**2)/(42*c*d**4) + 5*e**(7/2)*sqrt((c + d*x**2)/
(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(7*a**2*d**2 - 42*a*b*c*d + 39*b
**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(84*c**
(1/4)*d**(17/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.476266, size = 222, normalized size = 0.74 \[ \frac{(e x)^{7/2} \left (\frac{\sqrt{x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )+b^2 \left (-\left (195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right )\right )\right )}{d^4 \left (c+d x^2\right )}+\frac{5 i x \sqrt{\frac{c}{d x^2}+1} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{42 x^{7/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(7/2)*((Sqrt[x]*(-7*a^2*d^2*(5*c + 7*d*x^2) + 14*a*b*d*(15*c^2 + 21*c*d*x
^2 + 4*d^2*x^4) - b^2*(195*c^3 + 273*c^2*d*x^2 + 52*c*d^2*x^4 - 12*d^3*x^6)))/(d
^4*(c + d*x^2)) + ((5*I)*(39*b^2*c^2 - 42*a*b*c*d + 7*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^4)))/(42*x^(7/2)*Sqrt[c + d*x^2])

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Maple [B]  time = 0.062, size = 696, normalized size = 2.3 \[{\frac{{e}^{3}}{84\,x{d}^{5}} \left ( 35\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}{a}^{2}{d}^{3}-210\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}abc{d}^{2}+195\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}{b}^{2}{c}^{2}d+24\,{x}^{7}{b}^{2}{d}^{4}+35\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-210\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+195\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+112\,{x}^{5}ab{d}^{4}-104\,{x}^{5}{b}^{2}c{d}^{3}-98\,{x}^{3}{a}^{2}{d}^{4}+588\,{x}^{3}abc{d}^{3}-546\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}-70\,x{a}^{2}c{d}^{3}+420\,xab{c}^{2}{d}^{2}-390\,x{b}^{2}{c}^{3}d \right ) \sqrt{ex} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/84*(35*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(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)*(-c*d)^(1/2)*x^2*a^2*d^3-210*EllipticF(((d*x+(-c*d)^(
1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(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)*(-c*d)
^(1/2)*x^2*a*b*c*d^2+195*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2
^(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)*(-c*d)^(1/2)*x^2*b^2*c^2*d+24*x^7*b^2
*d^4+35*(-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^2-210*(-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*b*c^2*d+195*(-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+112*x^5*a*b*d^4-104*x^5
*b^2*c*d^3-98*x^3*a^2*d^4+588*x^3*a*b*c*d^3-546*x^3*b^2*c^2*d^2-70*x*a^2*c*d^3+4
20*x*a*b*c^2*d^2-390*x*b^2*c^3*d)*e^3/x*(e*x)^(1/2)/d^5/(d*x^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt{e x}}{{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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