3.1097 \(\int \frac{c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{8 \left (a+b x^2\right )^{5/4} (8 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2}}+\frac{2 \sqrt [4]{a+b x^2} (8 b c-9 a d)}{9 a^2 e^3 (e x)^{5/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}} \]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(9*a*e*(e*x)^(9/2)) + (2*(8*b*c - 9*a*d)*(a + b*x^2)^(1
/4))/(9*a^2*e^3*(e*x)^(5/2)) - (8*(8*b*c - 9*a*d)*(a + b*x^2)^(5/4))/(45*a^3*e^3
*(e*x)^(5/2))

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Rubi [A]  time = 0.169665, 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 \[ -\frac{8 \left (a+b x^2\right )^{5/4} (8 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2}}+\frac{2 \sqrt [4]{a+b x^2} (8 b c-9 a d)}{9 a^2 e^3 (e x)^{5/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(9*a*e*(e*x)^(9/2)) + (2*(8*b*c - 9*a*d)*(a + b*x^2)^(1
/4))/(9*a^2*e^3*(e*x)^(5/2)) - (8*(8*b*c - 9*a*d)*(a + b*x^2)^(5/4))/(45*a^3*e^3
*(e*x)^(5/2))

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Rubi in Sympy [A]  time = 17.6922, size = 99, normalized size = 0.95 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{9 a e \left (e x\right )^{\frac{9}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (9 a d - 8 b c\right )}{9 a^{2} e^{3} \left (e x\right )^{\frac{5}{2}}} + \frac{8 \left (a + b x^{2}\right )^{\frac{5}{4}} \left (9 a d - 8 b c\right )}{45 a^{3} e^{3} \left (e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(11/2)/(b*x**2+a)**(3/4),x)

[Out]

-2*c*(a + b*x**2)**(1/4)/(9*a*e*(e*x)**(9/2)) - 2*(a + b*x**2)**(1/4)*(9*a*d - 8
*b*c)/(9*a**2*e**3*(e*x)**(5/2)) + 8*(a + b*x**2)**(5/4)*(9*a*d - 8*b*c)/(45*a**
3*e**3*(e*x)**(5/2))

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Mathematica [A]  time = 0.100227, size = 72, normalized size = 0.69 \[ -\frac{2 \sqrt{e x} \sqrt [4]{a+b x^2} \left (a^2 \left (5 c+9 d x^2\right )-4 a b x^2 \left (2 c+9 d x^2\right )+32 b^2 c x^4\right )}{45 a^3 e^6 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*Sqrt[e*x]*(a + b*x^2)^(1/4)*(32*b^2*c*x^4 - 4*a*b*x^2*(2*c + 9*d*x^2) + a^2*
(5*c + 9*d*x^2)))/(45*a^3*e^6*x^5)

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Maple [A]  time = 0.01, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( -36\,{x}^{4}abd+32\,{b}^{2}c{x}^{4}+9\,{x}^{2}{a}^{2}d-8\,abc{x}^{2}+5\,{a}^{2}c \right ) }{45\,{a}^{3}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(3/4),x)

[Out]

-2/45*(b*x^2+a)^(1/4)*x*(-36*a*b*d*x^4+32*b^2*c*x^4+9*a^2*d*x^2-8*a*b*c*x^2+5*a^
2*c)/a^3/(e*x)^(11/2)

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Maxima [A]  time = 1.42257, size = 130, normalized size = 1.25 \[ \frac{2 \, d{\left (\frac{5 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b}{\sqrt{x}} - \frac{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}{x^{\frac{5}{2}}}\right )}}{5 \, a^{2} e^{\frac{11}{2}}} - \frac{2 \,{\left (\frac{45 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b^{2}}{\sqrt{x}} - \frac{18 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b}{x^{\frac{5}{2}}} + \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}}}{x^{\frac{9}{2}}}\right )} c}{45 \, a^{3} e^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(11/2)),x, algorithm="maxima")

[Out]

2/5*d*(5*(b*x^2 + a)^(1/4)*b/sqrt(x) - (b*x^2 + a)^(5/4)/x^(5/2))/(a^2*e^(11/2))
- 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))*c/(a^3*e^(11/2))

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Fricas [A]  time = 0.224322, size = 89, normalized size = 0.86 \[ -\frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 9 \, a b d\right )} x^{4} + 5 \, a^{2} c -{\left (8 \, a b c - 9 \, a^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{45 \, a^{3} e^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(11/2)),x, algorithm="fricas")

[Out]

-2/45*(4*(8*b^2*c - 9*a*b*d)*x^4 + 5*a^2*c - (8*a*b*c - 9*a^2*d)*x^2)*(b*x^2 + a
)^(1/4)*sqrt(e*x)/(a^3*e^6*x^5)

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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)**(11/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(11/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(11/2)), x)