3.1096 \(\int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.114783, 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 \[ \frac{2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.1354, size = 63, normalized size = 0.94 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{5 a e \left (e x\right )^{\frac{5}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (5 a d - 4 b c\right )}{5 a^{2} e^{3} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0657754, size = 44, normalized size = 0.66 \[ -\frac{2 x \sqrt [4]{a+b x^2} \left (a \left (c+5 d x^2\right )-4 b c x^2\right )}{5 a^2 (e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*(a + b*x^2)^(1/4)*(-4*b*c*x^2 + a*(c + 5*d*x^2)))/(5*a^2*(e*x)^(7/2))

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Maple [A]  time = 0.008, size = 39, normalized size = 0.6 \[ -{\frac{2\,x \left ( 5\,ad{x}^{2}-4\,c{x}^{2}b+ac \right ) }{5\,{a}^{2}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(b*x^2+a)^(1/4)*x*(5*a*d*x^2-4*b*c*x^2+a*c)/a^2/(e*x)^(7/2)

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Maxima [A]  time = 1.43469, size = 82, normalized size = 1.22 \[ \frac{2 \, c{\left (\frac{5 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b}{\sqrt{x}} - \frac{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}{x^{\frac{5}{2}}}\right )}}{5 \, a^{2} e^{\frac{7}{2}}} - \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} d}{a e^{\frac{7}{2}} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*c*(5*(b*x^2 + a)^(1/4)*b/sqrt(x) - (b*x^2 + a)^(5/4)/x^(5/2))/(a^2*e^(7/2))
- 2*(b*x^2 + a)^(1/4)*d/(a*e^(7/2)*sqrt(x))

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Fricas [A]  time = 0.226485, size = 58, normalized size = 0.87 \[ \frac{2 \,{\left ({\left (4 \, b c - 5 \, a d\right )} x^{2} - a c\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{5 \, a^{2} e^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(7/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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