3.1107 \(\int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.113632, antiderivative size = 67, 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 \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.0666, size = 61, normalized size = 0.91 \[ - \frac{2 c}{3 a e \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}}} + \frac{2 \sqrt{e x} \left (3 a d - 4 b c\right )}{3 a^{2} e^{3} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0670166, size = 45, normalized size = 0.67 \[ \frac{x \left (-2 a c+6 a d x^2-8 b c x^2\right )}{3 a^2 (e x)^{5/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-2*a*c - 8*b*c*x^2 + 6*a*d*x^2))/(3*a^2*(e*x)^(5/2)*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.008, size = 39, normalized size = 0.6 \[ -{\frac{2\,x \left ( -3\,ad{x}^{2}+4\,c{x}^{2}b+ac \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x*(-3*a*d*x^2+4*b*c*x^2+a*c)/(b*x^2+a)^(1/4)/a^2/(e*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222907, size = 57, normalized size = 0.85 \[ -\frac{2 \,{\left ({\left (4 \, b c - 3 \, a d\right )} x^{2} + a c\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x} a^{2} e^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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