3.1109 \(\int \frac{c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]

[Out]

(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(1/4)) - (2*(12*b*c - 11*a*d))/(11*a^2*e
^3*(e*x)^(7/2)*(a + b*x^2)^(1/4)) + (16*(12*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(33
*a^3*e^3*(e*x)^(7/2)) - (64*(12*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(231*a^4*e^3*(e
*x)^(7/2))

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Rubi [A]  time = 0.219422, 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 \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]

[Out]

(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(1/4)) - (2*(12*b*c - 11*a*d))/(11*a^2*e
^3*(e*x)^(7/2)*(a + b*x^2)^(1/4)) + (16*(12*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(33
*a^3*e^3*(e*x)^(7/2)) - (64*(12*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(231*a^4*e^3*(e
*x)^(7/2))

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Rubi in Sympy [A]  time = 23.334, size = 134, normalized size = 0.95 \[ - \frac{2 c}{11 a e \left (e x\right )^{\frac{11}{2}} \sqrt [4]{a + b x^{2}}} + \frac{2 \left (11 a d - 12 b c\right )}{11 a^{2} e^{3} \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}} - \frac{16 \left (a + b x^{2}\right )^{\frac{3}{4}} \left (11 a d - 12 b c\right )}{33 a^{3} e^{3} \left (e x\right )^{\frac{7}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{7}{4}} \left (11 a d - 12 b c\right )}{231 a^{4} e^{3} \left (e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

-2*c/(11*a*e*(e*x)**(11/2)*(a + b*x**2)**(1/4)) + 2*(11*a*d - 12*b*c)/(11*a**2*e
**3*(e*x)**(7/2)*(a + b*x**2)**(1/4)) - 16*(a + b*x**2)**(3/4)*(11*a*d - 12*b*c)
/(33*a**3*e**3*(e*x)**(7/2)) + 64*(a + b*x**2)**(7/4)*(11*a*d - 12*b*c)/(231*a**
4*e**3*(e*x)**(7/2))

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Mathematica [A]  time = 0.159672, size = 94, normalized size = 0.67 \[ \frac{\sqrt{e x} \left (-6 a^3 \left (7 c+11 d x^2\right )+8 a^2 b x^2 \left (9 c+22 d x^2\right )+64 a b^2 x^4 \left (11 d x^2-3 c\right )-768 b^3 c x^6\right )}{231 a^4 e^7 x^6 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]

[Out]

(Sqrt[e*x]*(-768*b^3*c*x^6 + 64*a*b^2*x^4*(-3*c + 11*d*x^2) - 6*a^3*(7*c + 11*d*
x^2) + 8*a^2*b*x^2*(9*c + 22*d*x^2)))/(231*a^4*e^7*x^6*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.009, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( -352\,a{b}^{2}d{x}^{6}+384\,{b}^{3}c{x}^{6}-88\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+33\,{a}^{3}d{x}^{2}-36\,{a}^{2}bc{x}^{2}+21\,c{a}^{3} \right ) }{231\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x)

[Out]

-2/231*x*(-352*a*b^2*d*x^6+384*b^3*c*x^6-88*a^2*b*d*x^4+96*a*b^2*c*x^4+33*a^3*d*
x^2-36*a^2*b*c*x^2+21*a^3*c)/(b*x^2+a)^(1/4)/a^4/(e*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)), x)

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Fricas [A]  time = 0.218447, size = 122, normalized size = 0.87 \[ -\frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{6} + 8 \,{\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} + 21 \, a^{3} c - 3 \,{\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{2}\right )}}{231 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x} a^{4} e^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="fricas")

[Out]

-2/231*(32*(12*b^3*c - 11*a*b^2*d)*x^6 + 8*(12*a*b^2*c - 11*a^2*b*d)*x^4 + 21*a^
3*c - 3*(12*a^2*b*c - 11*a^3*d)*x^2)/((b*x^2 + a)^(1/4)*sqrt(e*x)*a^4*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)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)), x)