3.1134 \(\int \frac{(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=155 \[ -\frac{3 e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} b^{5/2} \sqrt [4]{a+b x^2}}-\frac{e (e x)^{3/2} (2 b c-7 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{7/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[Out]

(2*(b*c - a*d)*(e*x)^(7/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((2*b*c - 7*a*d)*e*(e*
x)^(3/2))/(5*a*b^2*(a + b*x^2)^(1/4)) - (3*(2*b*c - 7*a*d)*e^2*(1 + a/(b*x^2))^(
1/4)*Sqrt[e*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*Sqrt[a]*b^(5/2)*(
a + b*x^2)^(1/4))

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Rubi [A]  time = 0.25518, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3 e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} b^{5/2} \sqrt [4]{a+b x^2}}-\frac{e (e x)^{3/2} (2 b c-7 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{7/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(2*(b*c - a*d)*(e*x)^(7/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((2*b*c - 7*a*d)*e*(e*
x)^(3/2))/(5*a*b^2*(a + b*x^2)^(1/4)) - (3*(2*b*c - 7*a*d)*e^2*(1 + a/(b*x^2))^(
1/4)*Sqrt[e*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*Sqrt[a]*b^(5/2)*(
a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d \left (e x\right )^{\frac{7}{2}}}{b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e \left (e x\right )^{\frac{3}{2}} \left (7 a d - 2 b c\right )}{5 b^{2} \left (a + b x^{2}\right )^{\frac{5}{4}}} - \frac{3 e^{2} \sqrt{e x} \left (\frac{7 a d}{2} - b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{5 b^{3} \sqrt [4]{a + b x^{2}}} + \frac{6 e^{2} \sqrt{e x} \left (\frac{7 a d}{2} - b c\right )}{5 b^{3} x \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

d*(e*x)**(7/2)/(b*e*(a + b*x**2)**(5/4)) + e*(e*x)**(3/2)*(7*a*d - 2*b*c)/(5*b**
2*(a + b*x**2)**(5/4)) - 3*e**2*sqrt(e*x)*(7*a*d/2 - b*c)*(a/(b*x**2) + 1)**(1/4
)*Integral((a*x**2/b + 1)**(-1/4), (x, 1/x))/(5*b**3*(a + b*x**2)**(1/4)) + 6*e*
*2*sqrt(e*x)*(7*a*d/2 - b*c)/(5*b**3*x*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.179201, size = 107, normalized size = 0.69 \[ \frac{2 e (e x)^{3/2} \left (-7 a^2 d+\left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (7 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+2 a b \left (c-4 d x^2\right )+3 b^2 c x^2\right )}{5 a b^2 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(2*e*(e*x)^(3/2)*(-7*a^2*d + 3*b^2*c*x^2 + 2*a*b*(c - 4*d*x^2) + (-2*b*c + 7*a*d
)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a
)]))/(5*a*b^2*(a + b*x^2)^(5/4))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

[Out]

int((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(9/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d e^{2} x^{4} + c e^{2} x^{2}\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(9/4),x, algorithm="fricas")

[Out]

integral((d*e^2*x^4 + c*e^2*x^2)*sqrt(e*x)/((b^2*x^4 + 2*a*b*x^2 + a^2)*(b*x^2 +
a)^(1/4)), 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)**(5/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(9/4), x)