3.1108 \(\int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{8 \left (a+b x^2\right )^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}} \]

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(1/4)) - (2*(8*b*c - 7*a*d))/(7*a^2*e^3*(e
*x)^(3/2)*(a + b*x^2)^(1/4)) + (8*(8*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(21*a^3*e^3
*(e*x)^(3/2))

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Rubi [A]  time = 0.164755, 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 \[ \frac{8 \left (a+b x^2\right )^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(1/4)) - (2*(8*b*c - 7*a*d))/(7*a^2*e^3*(e
*x)^(3/2)*(a + b*x^2)^(1/4)) + (8*(8*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(21*a^3*e^3
*(e*x)^(3/2))

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Rubi in Sympy [A]  time = 17.5452, size = 97, normalized size = 0.93 \[ - \frac{2 c}{7 a e \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}} + \frac{2 \left (7 a d - 8 b c\right )}{7 a^{2} e^{3} \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}}} - \frac{8 \left (a + b x^{2}\right )^{\frac{3}{4}} \left (7 a d - 8 b c\right )}{21 a^{3} e^{3} \left (e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c/(7*a*e*(e*x)**(7/2)*(a + b*x**2)**(1/4)) + 2*(7*a*d - 8*b*c)/(7*a**2*e**3*(
e*x)**(3/2)*(a + b*x**2)**(1/4)) - 8*(a + b*x**2)**(3/4)*(7*a*d - 8*b*c)/(21*a**
3*e**3*(e*x)**(3/2))

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Mathematica [A]  time = 0.105062, size = 71, normalized size = 0.68 \[ -\frac{2 \sqrt{e x} \left (a^2 \left (3 c+7 d x^2\right )+a b \left (28 d x^4-8 c x^2\right )-32 b^2 c x^4\right )}{21 a^3 e^5 x^4 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[e*x]*(-32*b^2*c*x^4 + a^2*(3*c + 7*d*x^2) + a*b*(-8*c*x^2 + 28*d*x^4)))
/(21*a^3*e^5*x^4*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.009, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( 28\,{x}^{4}abd-32\,{b}^{2}c{x}^{4}+7\,{x}^{2}{a}^{2}d-8\,abc{x}^{2}+3\,{a}^{2}c \right ) }{21\,{a}^{3}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*x*(28*a*b*d*x^4-32*b^2*c*x^4+7*a^2*d*x^2-8*a*b*c*x^2+3*a^2*c)/(b*x^2+a)^(1
/4)/a^3/(e*x)^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217903, size = 88, normalized size = 0.85 \[ \frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 7 \, a b d\right )} x^{4} - 3 \, a^{2} c +{\left (8 \, a b c - 7 \, a^{2} d\right )} x^{2}\right )}}{21 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x} a^{3} e^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(4*(8*b^2*c - 7*a*b*d)*x^4 - 3*a^2*c + (8*a*b*c - 7*a^2*d)*x^2)/((b*x^2 + a
)^(1/4)*sqrt(e*x)*a^3*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)**(9/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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