3.1107 \(\int \frac{c+d x^2}{(e x)^{5/2} (a+b x^2)^{5/4}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0305918, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {453, 264} \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{(4 b c-3 a d) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{5/4}} \, dx}{3 a e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (4 b c-3 a d) \sqrt{e x}}{3 a^2 e^3 \sqrt [4]{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0213481, size = 45, normalized size = 0.67 \[ \frac{x \left (-2 a c+6 a d x^2-8 b c x^2\right )}{3 a^2 (e x)^{5/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( -3\,ad{x}^{2}+4\,bc{x}^{2}+ac \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.87557, size = 124, normalized size = 1.85 \begin{align*} -\frac{2 \,{\left ({\left (4 \, b c - 3 \, a d\right )} x^{2} + a c\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{3 \,{\left (a^{2} b e^{3} x^{4} + a^{3} e^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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