3.2620 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{111311 x+66967}{39930 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{27 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}} \]

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (66967 + 111311*x)/(39930*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) +
(27*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10])

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Rubi [A]  time = 0.019763, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {98, 144, 54, 216} \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{111311 x+66967}{39930 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{27 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (66967 + 111311*x)/(39930*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) +
(27*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10])

Rule 98

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

Rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + 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
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

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{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x) \left (85+\frac{297 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{66967+111311 x}{39930 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{27}{20} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{66967+111311 x}{39930 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{27 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{10 \sqrt{5}}\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{66967+111311 x}{39930 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{27 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{10 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 0.15562, size = 143, normalized size = 1.7 \[ \frac{\sqrt{10-20 x} \sqrt{5 x+3} \left (21600 x^5-43740 x^4+79209 x^3+272474 x^2+678368 x+129582\right )-3993 \left (513 x^3+2538 x^2+936 x+334\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{79860 \sqrt{10} (1-2 x)^3}-\frac{250 \sqrt{\frac{2}{11}} (1-2 x)^{3/2} (3 x+2)^3 \, _2F_1\left (\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{5}{11} (1-2 x)\right )}{131769} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*(129582 + 678368*x + 272474*x^2 + 79209*x^3 - 43740*x^4 + 21600*x^5) - 3993*(33
4 + 936*x + 2538*x^2 + 513*x^3)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(79860*Sqrt[10]*(1 - 2*x)^3) - (250*Sqrt[2/1
1]*(1 - 2*x)^(3/2)*(2 + 3*x)^3*Hypergeometric2F1[3/2, 9/2, 11/2, (5*(1 - 2*x))/11])/131769

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Maple [B]  time = 0.012, size = 134, normalized size = 1.6 \begin{align*}{\frac{1}{798600\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 2156220\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-862488\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-754677\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5977040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+323433\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2485260\,x\sqrt{-10\,{x}^{2}-x+3}-661740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/798600*(1-2*x)^(1/2)*(2156220*10^(1/2)*arcsin(20/11*x+1/11)*x^3-862488*10^(1/2)*arcsin(20/11*x+1/11)*x^2-754
677*10^(1/2)*arcsin(20/11*x+1/11)*x+5977040*x^2*(-10*x^2-x+3)^(1/2)+323433*10^(1/2)*arcsin(20/11*x+1/11)+24852
60*x*(-10*x^2-x+3)^(1/2)-661740*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 2.03074, size = 105, normalized size = 1.25 \begin{align*} \frac{27}{200} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{74713 \, x}{19965 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{273689}{79860 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{343}{132 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/200*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 74713/19965*x/sqrt(-10*x^2 - x + 3) - 273689/79860/sqrt(-10*x^
2 - x + 3) - 343/132/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.57299, size = 308, normalized size = 3.67 \begin{align*} -\frac{107811 \, \sqrt{10}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 (298852 \, x^{2} + 124263 \, x - 33087\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{798600 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/798600*(107811*sqrt(10)*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 20*(298852*x^2 + 124263*x - 33087)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*
x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.74133, size = 159, normalized size = 1.89 \begin{align*} \frac{27}{100} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{66550 \, \sqrt{5 \, x + 3}} + \frac{49 \,{\left (244 \, \sqrt{5}{\left (5 \, x + 3\right )} - 957 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{199650 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{33275 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/100*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/66550*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) + 49/199650*(244*sqrt(5)*(5*x + 3) - 957*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/33
275*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))