3.2574 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4091 \sqrt{1-2 x}}{19965 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{3630 (5 x+3)^{3/2}}+\frac{49}{22 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(3630*(3 + 5*x)^(3/2)) - (4091*Sqrt[1 - 2*x])/(19
965*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0116792, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {89, 78, 37} \[ -\frac{4091 \sqrt{1-2 x}}{19965 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{3630 (5 x+3)^{3/2}}+\frac{49}{22 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^2/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(3630*(3 + 5*x)^(3/2)) - (4091*Sqrt[1 - 2*x])/(19
965*Sqrt[3 + 5*x])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{22} \int \frac{-\frac{617}{2}+99 x}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{3679 \sqrt{1-2 x}}{3630 (3+5 x)^{3/2}}+\frac{4091 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{3630}\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{3679 \sqrt{1-2 x}}{3630 (3+5 x)^{3/2}}-\frac{4091 \sqrt{1-2 x}}{19965 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0133729, size = 32, normalized size = 0.48 \[ \frac{2 \left (4091 x^2+4456 x+1196\right )}{3993 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^2/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*(1196 + 4456*x + 4091*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.4 \begin{align*}{\frac{8182\,{x}^{2}+8912\,x+2392}{3993} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x)

[Out]

2/3993*(4091*x^2+4456*x+1196)/(3+5*x)^(3/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.87475, size = 86, normalized size = 1.28 \begin{align*} \frac{8182 \, x}{19965 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{20014}{99825 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{825 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

8182/19965*x/sqrt(-10*x^2 - x + 3) + 20014/99825/sqrt(-10*x^2 - x + 3) - 2/825/(5*sqrt(-10*x^2 - x + 3)*x + 3*
sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.45822, size = 127, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (4091 \, x^{2} + 4456 \, x + 1196\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3993*(4091*x^2 + 4456*x + 1196)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{2}}{\left (1 - 2 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**2/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Integral((3*x + 2)**2/((1 - 2*x)**(3/2)*(5*x + 3)**(5/2)), x)

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Giac [B]  time = 2.67734, size = 205, normalized size = 3.06 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{1597200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{139 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{133100 \, \sqrt{5 \, x + 3}} - \frac{98 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6655 \,{\left (2 \, x - 1\right )}} + \frac{{\left (\frac{417 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{99825 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/1597200*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 139/133100*sqrt(10)*(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 98/6655*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/99825*(417
*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))^3