3.2319 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac{392 \sqrt{1-2 x} (3 x+2)^2}{825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1740 x+1243)}{11000}+\frac{1071 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (392*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(825*Sqrt[3 + 5*x]) + (7
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1243 + 1740*x))/11000 + (1071*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1000*Sqrt[10])

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Rubi [A]  time = 0.0302717, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {97, 150, 147, 54, 216} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac{392 \sqrt{1-2 x} (3 x+2)^2}{825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1740 x+1243)}{11000}+\frac{1071 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (392*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(825*Sqrt[3 + 5*x]) + (7
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1243 + 1740*x))/11000 + (1071*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1000*Sqrt[10])

Rule 97

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

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{(7-21 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{392 \sqrt{1-2 x} (2+3 x)^2}{825 \sqrt{3+5 x}}+\frac{4}{825} \int \frac{\left (357-\frac{3045 x}{2}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{392 \sqrt{1-2 x} (2+3 x)^2}{825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (1243+1740 x)}{11000}+\frac{1071 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{392 \sqrt{1-2 x} (2+3 x)^2}{825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (1243+1740 x)}{11000}+\frac{1071 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{392 \sqrt{1-2 x} (2+3 x)^2}{825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (1243+1740 x)}{11000}+\frac{1071 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0459841, size = 83, normalized size = 0.73 \[ \frac{-10 \left (178200 x^4+204930 x^3+3925 x^2-52336 x-11567\right )-35343 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{330000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-11567 - 52336*x + 3925*x^2 + 204930*x^3 + 178200*x^4) - 35343*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcSin[Sq
rt[5/11]*Sqrt[1 - 2*x]])/(330000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.01, size = 130, normalized size = 1.2 \begin{align*}{\frac{1}{660000} \left ( 883575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1782000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1060290\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+2940300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+318087\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1509400\,x\sqrt{-10\,{x}^{2}-x+3}+231340\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/660000*(883575*10^(1/2)*arcsin(20/11*x+1/11)*x^2+1782000*x^3*(-10*x^2-x+3)^(1/2)+1060290*10^(1/2)*arcsin(20/
11*x+1/11)*x+2940300*x^2*(-10*x^2-x+3)^(1/2)+318087*10^(1/2)*arcsin(20/11*x+1/11)+1509400*x*(-10*x^2-x+3)^(1/2
)+231340*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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

[Out]

Timed out

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Fricas [A]  time = 1.58691, size = 302, normalized size = 2.67 \begin{align*} -\frac{35343 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \,{\left (89100 \, x^{3} + 147015 \, x^{2} + 75470 \, x + 11567\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{660000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/660000*(35343*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) - 20*(89100*x^3 + 147015*x^2 + 75470*x + 11567)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x +
9)

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

[Out]

Timed out

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Giac [B]  time = 2.42056, size = 238, normalized size = 2.11 \begin{align*} \frac{27}{25000} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{1650000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{1071}{10000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{197 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{137500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{591 \, \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}}}{103125 \,{\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)^3*(1-2*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

27/25000*(4*sqrt(5)*(5*x + 3) - 3*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/1650000*sqrt(10)*(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1071/10000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 197/137500*
sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/103125*(591*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