3.1120 \(\int \frac{c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{64 \left (a+b x^2\right )^{5/4} (4 b c-3 a d)}{45 a^4 e^3 (e x)^{5/2}}+\frac{16 \sqrt [4]{a+b x^2} (4 b c-3 a d)}{9 a^3 e^3 (e x)^{5/2}}-\frac{2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}} \]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(3/4)) - (2*(4*b*c - 3*a*d))/(9*a^2*e^3*(e
*x)^(5/2)*(a + b*x^2)^(3/4)) + (16*(4*b*c - 3*a*d)*(a + b*x^2)^(1/4))/(9*a^3*e^3
*(e*x)^(5/2)) - (64*(4*b*c - 3*a*d)*(a + b*x^2)^(5/4))/(45*a^4*e^3*(e*x)^(5/2))

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Rubi [A]  time = 0.220013, 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 \[ -\frac{64 \left (a+b x^2\right )^{5/4} (4 b c-3 a d)}{45 a^4 e^3 (e x)^{5/2}}+\frac{16 \sqrt [4]{a+b x^2} (4 b c-3 a d)}{9 a^3 e^3 (e x)^{5/2}}-\frac{2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(7/4)),x]

[Out]

(-2*c)/(9*a*e*(e*x)^(9/2)*(a + b*x^2)^(3/4)) - (2*(4*b*c - 3*a*d))/(9*a^2*e^3*(e
*x)^(5/2)*(a + b*x^2)^(3/4)) + (16*(4*b*c - 3*a*d)*(a + b*x^2)^(1/4))/(9*a^3*e^3
*(e*x)^(5/2)) - (64*(4*b*c - 3*a*d)*(a + b*x^2)^(5/4))/(45*a^4*e^3*(e*x)^(5/2))

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Rubi in Sympy [A]  time = 22.7234, size = 134, normalized size = 0.95 \[ - \frac{2 c}{9 a e \left (e x\right )^{\frac{9}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (3 a d - 4 b c\right )}{9 a^{2} e^{3} \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{16 \sqrt [4]{a + b x^{2}} \left (3 a d - 4 b c\right )}{9 a^{3} e^{3} \left (e x\right )^{\frac{5}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{5}{4}} \left (3 a d - 4 b c\right )}{45 a^{4} e^{3} \left (e x\right )^{\frac{5}{2}}} \]

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)**(7/4),x)

[Out]

-2*c/(9*a*e*(e*x)**(9/2)*(a + b*x**2)**(3/4)) + 2*(3*a*d - 4*b*c)/(9*a**2*e**3*(
e*x)**(5/2)*(a + b*x**2)**(3/4)) - 16*(a + b*x**2)**(1/4)*(3*a*d - 4*b*c)/(9*a**
3*e**3*(e*x)**(5/2)) + 64*(a + b*x**2)**(5/4)*(3*a*d - 4*b*c)/(45*a**4*e**3*(e*x
)**(5/2))

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Mathematica [A]  time = 0.149288, size = 89, normalized size = 0.63 \[ -\frac{2 \sqrt{e x} \left (a^3 \left (5 c+9 d x^2\right )-12 a^2 b x^2 \left (c+6 d x^2\right )+96 a b^2 x^4 \left (c-d x^2\right )+128 b^3 c x^6\right )}{45 a^4 e^6 x^5 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(11/2)*(a + b*x^2)^(7/4)),x]

[Out]

(-2*Sqrt[e*x]*(128*b^3*c*x^6 + 96*a*b^2*x^4*(c - d*x^2) - 12*a^2*b*x^2*(c + 6*d*
x^2) + a^3*(5*c + 9*d*x^2)))/(45*a^4*e^6*x^5*(a + b*x^2)^(3/4))

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Maple [A]  time = 0.01, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( -96\,a{b}^{2}d{x}^{6}+128\,{b}^{3}c{x}^{6}-72\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+9\,{a}^{3}d{x}^{2}-12\,{a}^{2}bc{x}^{2}+5\,c{a}^{3} \right ) }{45\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(11/2)/(b*x^2+a)^(7/4),x)

[Out]

-2/45*x*(-96*a*b^2*d*x^6+128*b^3*c*x^6-72*a^2*b*d*x^4+96*a*b^2*c*x^4+9*a^3*d*x^2
-12*a^2*b*c*x^2+5*a^3*c)/(b*x^2+a)^(3/4)/a^4/(e*x)^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(11/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(11/2)), x)

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Fricas [A]  time = 0.251963, size = 142, normalized size = 1.01 \[ -\frac{2 \,{\left (32 \,{\left (4 \, b^{3} c - 3 \, a b^{2} d\right )} x^{6} + 24 \,{\left (4 \, a b^{2} c - 3 \, a^{2} b d\right )} x^{4} + 5 \, a^{3} c - 3 \,{\left (4 \, a^{2} b c - 3 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{45 \,{\left (a^{4} b e^{6} x^{7} + a^{5} e^{6} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(11/2)),x, algorithm="fricas")

[Out]

-2/45*(32*(4*b^3*c - 3*a*b^2*d)*x^6 + 24*(4*a*b^2*c - 3*a^2*b*d)*x^4 + 5*a^3*c -
3*(4*a^2*b*c - 3*a^3*d)*x^2)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(a^4*b*e^6*x^7 + a^5*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)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(11/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(11/2)), x)