3.1128 \(\int \frac{c+d x^2}{\sqrt{e x} (a+b x^2)^{9/4}} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 \sqrt{e x} (a d+4 b c)}{5 a^2 b e \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{e x} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[Out]

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

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Rubi [A]  time = 0.0328255, antiderivative size = 79, 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 = {457, 264} \[ \frac{2 \sqrt{e x} (a d+4 b c)}{5 a^2 b e \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{e x} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(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}{\sqrt{e x} \left (a+b x^2\right )^{9/4}} \, dx &=\frac{2 (b c-a d) \sqrt{e x}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac{\left (2 \left (2 b c+\frac{a d}{2}\right )\right ) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{5/4}} \, dx}{5 a b}\\ &=\frac{2 (b c-a d) \sqrt{e x}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac{2 (4 b c+a d) \sqrt{e x}}{5 a^2 b e \sqrt [4]{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0460489, size = 44, normalized size = 0.56 \[ \frac{2 x \left (5 a c+a d x^2+4 b c x^2\right )}{5 a^2 \sqrt{e x} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 0.5 \begin{align*}{\frac{2\,x \left ( ad{x}^{2}+4\,bc{x}^{2}+5\,ac \right ) }{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}{\frac{1}{\sqrt{ex}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(9/4),x)

[Out]

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

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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{9}{4}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(9/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.00099, size = 136, normalized size = 1.72 \begin{align*} \frac{2 \,{\left ({\left (4 \, b c + a d\right )} x^{2} + 5 \, a c\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{5 \,{\left (a^{2} b^{2} e x^{4} + 2 \, a^{3} b e x^{2} + a^{4} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(9/4),x, algorithm="fricas")

[Out]

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

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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)**(1/2)/(b*x**2+a)**(9/4),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{9}{4}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(9/4),x, algorithm="giac")

[Out]

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