3.1131 \(\int \frac{c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac{64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]

[Out]

(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(5/4)) - (2*(16*b*c - 11*a*d))/(55*a^2*e
^3*(e*x)^(7/2)*(a + b*x^2)^(5/4)) - (24*(16*b*c - 11*a*d))/(55*a^3*e^3*(e*x)^(7/
2)*(a + b*x^2)^(1/4)) + (64*(16*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(55*a^4*e^3*(e*
x)^(7/2)) - (256*(16*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(385*a^5*e^3*(e*x)^(7/2))

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Rubi [A]  time = 0.277489, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac{64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]

[Out]

(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(5/4)) - (2*(16*b*c - 11*a*d))/(55*a^2*e
^3*(e*x)^(7/2)*(a + b*x^2)^(5/4)) - (24*(16*b*c - 11*a*d))/(55*a^3*e^3*(e*x)^(7/
2)*(a + b*x^2)^(1/4)) + (64*(16*b*c - 11*a*d)*(a + b*x^2)^(3/4))/(55*a^4*e^3*(e*
x)^(7/2)) - (256*(16*b*c - 11*a*d)*(a + b*x^2)^(7/4))/(385*a^5*e^3*(e*x)^(7/2))

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Rubi in Sympy [A]  time = 29.0576, size = 170, normalized size = 0.96 \[ - \frac{2 c}{11 a e \left (e x\right )^{\frac{11}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \left (11 a d - 16 b c\right )}{55 a^{2} e^{3} \left (e x\right )^{\frac{7}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{24 \left (11 a d - 16 b c\right )}{55 a^{3} e^{3} \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{3}{4}} \left (11 a d - 16 b c\right )}{55 a^{4} e^{3} \left (e x\right )^{\frac{7}{2}}} + \frac{256 \left (a + b x^{2}\right )^{\frac{7}{4}} \left (11 a d - 16 b c\right )}{385 a^{5} e^{3} \left (e x\right )^{\frac{7}{2}}} \]

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)**(9/4),x)

[Out]

-2*c/(11*a*e*(e*x)**(11/2)*(a + b*x**2)**(5/4)) + 2*(11*a*d - 16*b*c)/(55*a**2*e
**3*(e*x)**(7/2)*(a + b*x**2)**(5/4)) + 24*(11*a*d - 16*b*c)/(55*a**3*e**3*(e*x)
**(7/2)*(a + b*x**2)**(1/4)) - 64*(a + b*x**2)**(3/4)*(11*a*d - 16*b*c)/(55*a**4
*e**3*(e*x)**(7/2)) + 256*(a + b*x**2)**(7/4)*(11*a*d - 16*b*c)/(385*a**5*e**3*(
e*x)**(7/2))

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Mathematica [A]  time = 0.209614, size = 115, normalized size = 0.65 \[ -\frac{2 \sqrt{e x} \left (5 a^4 \left (7 c+11 d x^2\right )-20 a^3 b x^2 \left (4 c+11 d x^2\right )+160 a^2 b^2 x^4 \left (2 c-11 d x^2\right )+128 a b^3 x^6 \left (20 c-11 d x^2\right )+2048 b^4 c x^8\right )}{385 a^5 e^7 x^6 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]

[Out]

(-2*Sqrt[e*x]*(2048*b^4*c*x^8 + 160*a^2*b^2*x^4*(2*c - 11*d*x^2) + 128*a*b^3*x^6
*(20*c - 11*d*x^2) - 20*a^3*b*x^2*(4*c + 11*d*x^2) + 5*a^4*(7*c + 11*d*x^2)))/(3
85*a^5*e^7*x^6*(a + b*x^2)^(5/4))

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Maple [A]  time = 0.01, size = 110, normalized size = 0.6 \[ -{\frac{2\,x \left ( -1408\,a{b}^{3}d{x}^{8}+2048\,{b}^{4}c{x}^{8}-1760\,{a}^{2}{b}^{2}d{x}^{6}+2560\,a{b}^{3}c{x}^{6}-220\,{a}^{3}bd{x}^{4}+320\,{a}^{2}{b}^{2}c{x}^{4}+55\,{a}^{4}d{x}^{2}-80\,{a}^{3}bc{x}^{2}+35\,c{a}^{4} \right ) }{385\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x)

[Out]

-2/385*x*(-1408*a*b^3*d*x^8+2048*b^4*c*x^8-1760*a^2*b^2*d*x^6+2560*a*b^3*c*x^6-2
20*a^3*b*d*x^4+320*a^2*b^2*c*x^4+55*a^4*d*x^2-80*a^3*b*c*x^2+35*a^4*c)/(b*x^2+a)
^(5/4)/a^5/(e*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)), x)

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Fricas [A]  time = 0.24504, size = 174, normalized size = 0.98 \[ -\frac{2 \,{\left (128 \,{\left (16 \, b^{4} c - 11 \, a b^{3} d\right )} x^{8} + 160 \,{\left (16 \, a b^{3} c - 11 \, a^{2} b^{2} d\right )} x^{6} + 35 \, a^{4} c + 20 \,{\left (16 \, a^{2} b^{2} c - 11 \, a^{3} b d\right )} x^{4} - 5 \,{\left (16 \, a^{3} b c - 11 \, a^{4} d\right )} x^{2}\right )}}{385 \,{\left (a^{5} b e^{6} x^{7} + a^{6} e^{6} x^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)),x, algorithm="fricas")

[Out]

-2/385*(128*(16*b^4*c - 11*a*b^3*d)*x^8 + 160*(16*a*b^3*c - 11*a^2*b^2*d)*x^6 +
35*a^4*c + 20*(16*a^2*b^2*c - 11*a^3*b*d)*x^4 - 5*(16*a^3*b*c - 11*a^4*d)*x^2)/(
(a^5*b*e^6*x^7 + a^6*e^6*x^5)*(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)**(13/2)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)), x)