3.1129 \(\int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{8 \sqrt{e x} (8 b c-3 a d)}{15 a^3 e^3 \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{e x} (8 b c-3 a d)}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}} \]

[Out]

(-2*c)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(5/4)) - (2*(8*b*c - 3*a*d)*Sqrt[e*x])/(15
*a^2*e^3*(a + b*x^2)^(5/4)) - (8*(8*b*c - 3*a*d)*Sqrt[e*x])/(15*a^3*e^3*(a + b*x
^2)^(1/4))

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Rubi [A]  time = 0.169736, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{8 \sqrt{e x} (8 b c-3 a d)}{15 a^3 e^3 \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{e x} (8 b c-3 a d)}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c)/(3*a*e*(e*x)^(3/2)*(a + b*x^2)^(5/4)) - (2*(8*b*c - 3*a*d)*Sqrt[e*x])/(15
*a^2*e^3*(a + b*x^2)^(5/4)) - (8*(8*b*c - 3*a*d)*Sqrt[e*x])/(15*a^3*e^3*(a + b*x
^2)^(1/4))

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Rubi in Sympy [A]  time = 17.435, size = 99, normalized size = 0.95 \[ - \frac{2 c}{3 a e \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \sqrt{e x} \left (3 a d - 8 b c\right )}{15 a^{2} e^{3} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{8 \sqrt{e x} \left (3 a d - 8 b c\right )}{15 a^{3} e^{3} \sqrt [4]{a + b x^{2}}} \]

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)**(9/4),x)

[Out]

-2*c/(3*a*e*(e*x)**(3/2)*(a + b*x**2)**(5/4)) + 2*sqrt(e*x)*(3*a*d - 8*b*c)/(15*
a**2*e**3*(a + b*x**2)**(5/4)) + 8*sqrt(e*x)*(3*a*d - 8*b*c)/(15*a**3*e**3*(a +
b*x**2)**(1/4))

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Mathematica [A]  time = 0.0903344, size = 65, normalized size = 0.62 \[ \frac{x \left (-10 a^2 \left (c-3 d x^2\right )+a b \left (24 d x^4-80 c x^2\right )-64 b^2 c x^4\right )}{15 a^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-64*b^2*c*x^4 - 10*a^2*(c - 3*d*x^2) + a*b*(-80*c*x^2 + 24*d*x^4)))/(15*a^3*
(e*x)^(5/2)*(a + b*x^2)^(5/4))

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Maple [A]  time = 0.01, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( -12\,{x}^{4}abd+32\,{b}^{2}c{x}^{4}-15\,{x}^{2}{a}^{2}d+40\,abc{x}^{2}+5\,{a}^{2}c \right ) }{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*x*(-12*a*b*d*x^4+32*b^2*c*x^4-15*a^2*d*x^2+40*a*b*c*x^2+5*a^2*c)/(b*x^2+a)
^(5/4)/a^3/(e*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24599, size = 107, normalized size = 1.03 \[ -\frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 3 \, a b d\right )} x^{4} + 5 \, a^{2} c + 5 \,{\left (8 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )}}{15 \,{\left (a^{3} b e^{2} x^{3} + a^{4} e^{2} x\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(4*(8*b^2*c - 3*a*b*d)*x^4 + 5*a^2*c + 5*(8*a*b*c - 3*a^2*d)*x^2)/((a^3*b*
e^2*x^3 + a^4*e^2*x)*(b*x^2 + a)^(1/4)*sqrt(e*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)**(9/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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