3.1119 \(\int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.170531, 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 \[ \frac{8 \sqrt [4]{a+b x^2} (8 b c-5 a d)}{15 a^3 e^3 \sqrt{e x}}-\frac{2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt{e x} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.1046, size = 97, normalized size = 0.93 \[ - \frac{2 c}{5 a e \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (5 a d - 8 b c\right )}{15 a^{2} e^{3} \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{8 \sqrt [4]{a + b x^{2}} \left (5 a d - 8 b c\right )}{15 a^{3} e^{3} \sqrt{e x}} \]

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)**(7/4),x)

[Out]

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

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Mathematica [A]  time = 0.100183, size = 66, normalized size = 0.63 \[ \frac{x \left (-6 a^2 \left (c+5 d x^2\right )+8 a b x^2 \left (6 c-5 d x^2\right )+64 b^2 c x^4\right )}{15 a^3 (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(64*b^2*c*x^4 + 8*a*b*x^2*(6*c - 5*d*x^2) - 6*a^2*(c + 5*d*x^2)))/(15*a^3*(e*
x)^(7/2)*(a + b*x^2)^(3/4))

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Maple [A]  time = 0.009, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( 20\,{x}^{4}abd-32\,{b}^{2}c{x}^{4}+15\,{x}^{2}{a}^{2}d-24\,abc{x}^{2}+3\,{a}^{2}c \right ) }{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*x*(20*a*b*d*x^4-32*b^2*c*x^4+15*a^2*d*x^2-24*a*b*c*x^2+3*a^2*c)/(b*x^2+a)^
(3/4)/a^3/(e*x)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237579, size = 109, normalized size = 1.05 \[ \frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 5 \, a b d\right )} x^{4} - 3 \, a^{2} c + 3 \,{\left (8 \, a b c - 5 \, a^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{15 \,{\left (a^{3} b e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(4*(8*b^2*c - 5*a*b*d)*x^4 - 3*a^2*c + 3*(8*a*b*c - 5*a^2*d)*x^2)*(b*x^2 +
a)^(1/4)*sqrt(e*x)/(a^3*b*e^4*x^5 + a^4*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)**(7/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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