3.1098 \(\int \frac{c+d x^2}{(e x)^{15/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=141 \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(13*a*e*(e*x)^(13/2)) + (2*(12*b*c - 13*a*d)*(a + b*x^2
)^(1/4))/(13*a^2*e^3*(e*x)^(9/2)) - (16*(12*b*c - 13*a*d)*(a + b*x^2)^(5/4))/(65
*a^3*e^3*(e*x)^(9/2)) + (64*(12*b*c - 13*a*d)*(a + b*x^2)^(9/4))/(585*a^4*e^3*(e
*x)^(9/2))

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Rubi [A]  time = 0.222218, 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 \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(13*a*e*(e*x)^(13/2)) + (2*(12*b*c - 13*a*d)*(a + b*x^2
)^(1/4))/(13*a^2*e^3*(e*x)^(9/2)) - (16*(12*b*c - 13*a*d)*(a + b*x^2)^(5/4))/(65
*a^3*e^3*(e*x)^(9/2)) + (64*(12*b*c - 13*a*d)*(a + b*x^2)^(9/4))/(585*a^4*e^3*(e
*x)^(9/2))

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Rubi in Sympy [A]  time = 23.4163, size = 136, normalized size = 0.96 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{13 a e \left (e x\right )^{\frac{13}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (13 a d - 12 b c\right )}{13 a^{2} e^{3} \left (e x\right )^{\frac{9}{2}}} + \frac{16 \left (a + b x^{2}\right )^{\frac{5}{4}} \left (13 a d - 12 b c\right )}{65 a^{3} e^{3} \left (e x\right )^{\frac{9}{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{9}{4}} \left (13 a d - 12 b c\right )}{585 a^{4} e^{3} \left (e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(15/2)/(b*x**2+a)**(3/4),x)

[Out]

-2*c*(a + b*x**2)**(1/4)/(13*a*e*(e*x)**(13/2)) - 2*(a + b*x**2)**(1/4)*(13*a*d
- 12*b*c)/(13*a**2*e**3*(e*x)**(9/2)) + 16*(a + b*x**2)**(5/4)*(13*a*d - 12*b*c)
/(65*a**3*e**3*(e*x)**(9/2)) - 64*(a + b*x**2)**(9/4)*(13*a*d - 12*b*c)/(585*a**
4*e**3*(e*x)**(9/2))

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Mathematica [A]  time = 0.119092, size = 94, normalized size = 0.67 \[ -\frac{2 \sqrt{e x} \sqrt [4]{a+b x^2} \left (5 a^3 \left (9 c+13 d x^2\right )-4 a^2 b x^2 \left (15 c+26 d x^2\right )+32 a b^2 x^4 \left (3 c+13 d x^2\right )-384 b^3 c x^6\right )}{585 a^4 e^8 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*Sqrt[e*x]*(a + b*x^2)^(1/4)*(-384*b^3*c*x^6 + 32*a*b^2*x^4*(3*c + 13*d*x^2)
+ 5*a^3*(9*c + 13*d*x^2) - 4*a^2*b*x^2*(15*c + 26*d*x^2)))/(585*a^4*e^8*x^7)

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Maple [A]  time = 0.009, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( 416\,a{b}^{2}d{x}^{6}-384\,{b}^{3}c{x}^{6}-104\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+65\,{a}^{3}d{x}^{2}-60\,{a}^{2}bc{x}^{2}+45\,c{a}^{3} \right ) }{585\,{a}^{4}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x)

[Out]

-2/585*(b*x^2+a)^(1/4)*x*(416*a*b^2*d*x^6-384*b^3*c*x^6-104*a^2*b*d*x^4+96*a*b^2
*c*x^4+65*a^3*d*x^2-60*a^2*b*c*x^2+45*a^3*c)/a^4/(e*x)^(15/2)

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Maxima [A]  time = 1.43236, size = 176, normalized size = 1.25 \[ -\frac{2 \,{\left (\frac{45 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b^{2}}{\sqrt{x}} - \frac{18 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b}{x^{\frac{5}{2}}} + \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}}}{x^{\frac{9}{2}}}\right )} d}{45 \, a^{3} e^{\frac{15}{2}}} + \frac{2 \,{\left (\frac{195 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b^{3}}{\sqrt{x}} - \frac{117 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b^{2}}{x^{\frac{5}{2}}} + \frac{65 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}} b}{x^{\frac{9}{2}}} - \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{13}{4}}}{x^{\frac{13}{2}}}\right )} c}{195 \, a^{4} e^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),x, algorithm="maxima")

[Out]

-2/45*(45*(b*x^2 + a)^(1/4)*b^2/sqrt(x) - 18*(b*x^2 + a)^(5/4)*b/x^(5/2) + 5*(b*
x^2 + a)^(9/4)/x^(9/2))*d/(a^3*e^(15/2)) + 2/195*(195*(b*x^2 + a)^(1/4)*b^3/sqrt
(x) - 117*(b*x^2 + a)^(5/4)*b^2/x^(5/2) + 65*(b*x^2 + a)^(9/4)*b/x^(9/2) - 15*(b
*x^2 + a)^(13/4)/x^(13/2))*c/(a^4*e^(15/2))

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Fricas [A]  time = 0.222813, size = 122, normalized size = 0.87 \[ \frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{6} - 8 \,{\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{4} - 45 \, a^{3} c + 5 \,{\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{585 \, a^{4} e^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),x, algorithm="fricas")

[Out]

2/585*(32*(12*b^3*c - 13*a*b^2*d)*x^6 - 8*(12*a*b^2*c - 13*a^2*b*d)*x^4 - 45*a^3
*c + 5*(12*a^2*b*c - 13*a^3*d)*x^2)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(a^4*e^8*x^7)

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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)**(15/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)), x)