3.1122 \(\int \frac{(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{e \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-5 a d)}{3 a b^2}+\frac{2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.334732, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{e \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-5 a d)}{3 a b^2}+\frac{2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.0309, size = 124, normalized size = 0.82 \[ \frac{d \left (e x\right )^{\frac{5}{2}}}{b e \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{e \sqrt{e x} \left (5 a d - 2 b c\right )}{3 b^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (\frac{5 a d}{2} - b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} b^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)

[Out]

d*(e*x)**(5/2)/(b*e*(a + b*x**2)**(3/4)) + e*sqrt(e*x)*(5*a*d - 2*b*c)/(3*b**2*(
a + b*x**2)**(3/4)) + 2*(e*x)**(3/2)*(5*a*d/2 - b*c)*(a/(b*x**2) + 1)**(3/4)*ell
iptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(3*sqrt(a)*b**(3/2)*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.112707, size = 85, normalized size = 0.56 \[ \frac{e \sqrt{e x} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} (2 b c-5 a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+5 a d-2 b c+3 b d x^2\right )}{3 b^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]

[Out]

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

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(7/4),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(7/4),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(7/4),x, algorithm="giac")

[Out]

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