3.1130 \(\int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=141 \[ \frac{64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(5/4)) - (2*(12*b*c - 7*a*d))/(35*a^2*e^3*
(e*x)^(3/2)*(a + b*x^2)^(5/4)) - (16*(12*b*c - 7*a*d))/(35*a^3*e^3*(e*x)^(3/2)*(
a + b*x^2)^(1/4)) + (64*(12*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(105*a^4*e^3*(e*x)^(
3/2))

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Rubi [A]  time = 0.21835, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(5/4)) - (2*(12*b*c - 7*a*d))/(35*a^2*e^3*
(e*x)^(3/2)*(a + b*x^2)^(5/4)) - (16*(12*b*c - 7*a*d))/(35*a^3*e^3*(e*x)^(3/2)*(
a + b*x^2)^(1/4)) + (64*(12*b*c - 7*a*d)*(a + b*x^2)^(3/4))/(105*a^4*e^3*(e*x)^(
3/2))

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Rubi in Sympy [A]  time = 22.8129, size = 133, normalized size = 0.94 \[ - \frac{2 c}{7 a e \left (e x\right )^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \left (7 a d - 12 b c\right )}{35 a^{2} e^{3} \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{16 \left (7 a d - 12 b c\right )}{35 a^{3} e^{3} \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{3}{4}} \left (7 a d - 12 b c\right )}{105 a^{4} e^{3} \left (e x\right )^{\frac{3}{2}}} \]

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)**(9/4),x)

[Out]

-2*c/(7*a*e*(e*x)**(7/2)*(a + b*x**2)**(5/4)) + 2*(7*a*d - 12*b*c)/(35*a**2*e**3
*(e*x)**(3/2)*(a + b*x**2)**(5/4)) + 16*(7*a*d - 12*b*c)/(35*a**3*e**3*(e*x)**(3
/2)*(a + b*x**2)**(1/4)) - 64*(a + b*x**2)**(3/4)*(7*a*d - 12*b*c)/(105*a**4*e**
3*(e*x)**(3/2))

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Mathematica [A]  time = 0.148512, size = 94, normalized size = 0.67 \[ \frac{\sqrt{e x} \left (-10 a^3 \left (3 c+7 d x^2\right )+40 a^2 b x^2 \left (3 c-14 d x^2\right )+64 a b^2 x^4 \left (15 c-7 d x^2\right )+768 b^3 c x^6\right )}{105 a^4 e^5 x^4 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[e*x]*(768*b^3*c*x^6 + 40*a^2*b*x^2*(3*c - 14*d*x^2) + 64*a*b^2*x^4*(15*c -
7*d*x^2) - 10*a^3*(3*c + 7*d*x^2)))/(105*a^4*e^5*x^4*(a + b*x^2)^(5/4))

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Maple [A]  time = 0.009, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( 224\,a{b}^{2}d{x}^{6}-384\,{b}^{3}c{x}^{6}+280\,{a}^{2}bd{x}^{4}-480\,a{b}^{2}c{x}^{4}+35\,{a}^{3}d{x}^{2}-60\,{a}^{2}bc{x}^{2}+15\,c{a}^{3} \right ) }{105\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*x*(224*a*b^2*d*x^6-384*b^3*c*x^6+280*a^2*b*d*x^4-480*a*b^2*c*x^4+35*a^3*d
*x^2-60*a^2*b*c*x^2+15*a^3*c)/(b*x^2+a)^(5/4)/a^4/(e*x)^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245767, size = 142, normalized size = 1.01 \[ \frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 7 \, a b^{2} d\right )} x^{6} + 40 \,{\left (12 \, a b^{2} c - 7 \, a^{2} b d\right )} x^{4} - 15 \, a^{3} c + 5 \,{\left (12 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )}}{105 \,{\left (a^{4} b e^{4} x^{5} + a^{5} e^{4} x^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(32*(12*b^3*c - 7*a*b^2*d)*x^6 + 40*(12*a*b^2*c - 7*a^2*b*d)*x^4 - 15*a^3*
c + 5*(12*a^2*b*c - 7*a^3*d)*x^2)/((a^4*b*e^4*x^5 + a^5*e^4*x^3)*(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)**(9/2)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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