3.1118 \(\int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.117032, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.8793, size = 58, normalized size = 0.89 \[ - \frac{2 c}{a e \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (a d - 4 b c\right )}{3 a^{2} e^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c/(a*e*sqrt(e*x)*(a + b*x**2)**(3/4)) + 2*(e*x)**(3/2)*(a*d - 4*b*c)/(3*a**2*
e**3*(a + b*x**2)**(3/4))

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Mathematica [A]  time = 0.0570357, size = 44, normalized size = 0.68 \[ \frac{2 x \left (-3 a c+a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-3*a*c - 4*b*c*x^2 + a*d*x^2))/(3*a^2*(e*x)^(3/2)*(a + b*x^2)^(3/4))

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Maple [A]  time = 0.009, size = 40, normalized size = 0.6 \[ -{\frac{2\,x \left ( -ad{x}^{2}+4\,c{x}^{2}b+3\,ac \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x*(-a*d*x^2+4*b*c*x^2+3*a*c)/(b*x^2+a)^(3/4)/a^2/(e*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242057, size = 76, normalized size = 1.17 \[ -\frac{2 \,{\left ({\left (4 \, b c - a d\right )} x^{2} + 3 \, a c\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{3 \,{\left (a^{2} b e^{2} x^{3} + a^{3} e^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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