3.2571 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^3}{1815 (5 x+3)^{3/2}}-\frac{4487 \sqrt{1-2 x} (3 x+2)^2}{99825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1078860 x+2571547)}{5324000}-\frac{111321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) -
(4487*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(99825*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2571547 + 1078860*x)
)/5324000 - (111321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*Sqrt[10])

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Rubi [A]  time = 0.0409921, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^3}{1815 (5 x+3)^{3/2}}-\frac{4487 \sqrt{1-2 x} (3 x+2)^2}{99825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1078860 x+2571547)}{5324000}-\frac{111321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) -
(4487*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(99825*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2571547 + 1078860*x)
)/5324000 - (111321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*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 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{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{11} \int \frac{(2+3 x)^3 \left (145+\frac{519 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2 \int \frac{(2+3 x)^2 \left (7868+\frac{53949 x}{4}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4487 \sqrt{1-2 x} (2+3 x)^2}{99825 \sqrt{3+5 x}}-\frac{4 \int \frac{(2+3 x) \left (\frac{566517}{4}+\frac{1888005 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{99825}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4487 \sqrt{1-2 x} (2+3 x)^2}{99825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2571547+1078860 x)}{5324000}-\frac{111321 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{8000}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4487 \sqrt{1-2 x} (2+3 x)^2}{99825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2571547+1078860 x)}{5324000}-\frac{111321 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{4000 \sqrt{5}}\\ &=-\frac{107 \sqrt{1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4487 \sqrt{1-2 x} (2+3 x)^2}{99825 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2571547+1078860 x)}{5324000}-\frac{111321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{4000 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 1.20279, size = 280, normalized size = 1.97 \[ \frac{173 \left (-1320000 (3 x+2)^3 (1-2 x)^{7/2} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},2,2,\frac{7}{2}\right \},\left \{1,1,\frac{9}{2}\right \},\frac{5}{11} (1-2 x)\right )-1050000 (x+3) \left (6 x^2+x-2\right )^2 (1-2 x)^{5/2} \, _2F_1\left (\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{5}{11} (1-2 x)\right )+77 \sqrt{55} \left (\sqrt{10-20 x} \sqrt{5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )-27951 \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )\right )}{2635380000 \sqrt{22} (1-2 x)^3}+\frac{189 \left (\frac{10 \sqrt{1-2 x} \left (49005 x^2+60010 x+18373\right )}{(5 x+3)^{3/2}}+29403 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{4840000}-\frac{3 (3 x+2)^4}{20 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(2 + 3*x)^4)/(20*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (189*((10*Sqrt[1 - 2*x]*(18373 + 60010*x + 49005*x^2))/(
3 + 5*x)^(3/2) + 29403*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]))/4840000 + (173*(77*Sqrt[55]*(Sqrt[10 - 20*x
]*Sqrt[3 + 5*x]*(1669914 + 2053496*x + 1229303*x^2 - 400032*x^3 + 28080*x^4 + 43200*x^5) - 27951*(374 + 1296*x
+ 513*x^2 + 108*x^3)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]) - 1050000*(1 - 2*x)^(5/2)*(3 + x)*(-2 + x + 6*x^2)^2*H
ypergeometric2F1[3/2, 9/2, 11/2, (5*(1 - 2*x))/11] - 1320000*(1 - 2*x)^(7/2)*(2 + 3*x)^3*HypergeometricPFQ[{1/
2, 2, 2, 7/2}, {1, 1, 9/2}, (5*(1 - 2*x))/11]))/(2635380000*Sqrt[22]*(1 - 2*x)^3)

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Maple [A]  time = 0.015, size = 168, normalized size = 1.2 \begin{align*} -{\frac{1}{638880000\,x-319440000}\sqrt{1-2\,x} \left ( 22225237650\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-3881196000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-22575623400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-5334057036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+12242129500\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4000542777\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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)^5/(1-2*x)^(3/2)/(3+5*x)^(5/2),x)

[Out]

-1/319440000*(1-2*x)^(1/2)*(22225237650*10^(1/2)*arcsin(20/11*x+1/11)*x^3-3881196000*x^4*(-10*x^2-x+3)^(1/2)+1
5557666355*10^(1/2)*arcsin(20/11*x+1/11)*x^2-22575623400*x^3*(-10*x^2-x+3)^(1/2)-5334057036*10^(1/2)*arcsin(20
/11*x+1/11)*x+12242129500*x^2*(-10*x^2-x+3)^(1/2)-4000542777*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*
x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(2*x-1)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.81357, size = 151, normalized size = 1.06 \begin{align*} -\frac{243 \, x^{3}}{100 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{111321}{80000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{25353 \, x^{2}}{2000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1219513649 \, x}{79860000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5270823773}{399300000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{103125 \,{\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)^5/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-243/100*x^3/sqrt(-10*x^2 - x + 3) - 111321/80000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 25353/2000*x^2/sqrt
(-10*x^2 - x + 3) + 1219513649/79860000*x/sqrt(-10*x^2 - x + 3) + 5270823773/399300000/sqrt(-10*x^2 - x + 3) -
2/103125/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.56063, size = 379, normalized size = 2.67 \begin{align*} \frac{444504753 \, \sqrt{10}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, 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 (194059800 \, x^{4} + 1128781170 \, x^{3} - 612106475 \, x^{2} - 1785872944 \, x - 632498543\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{319440000 \,{\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)^5/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

1/319440000*(444504753*sqrt(10)*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqr
t(-2*x + 1)/(10*x^2 + x - 3)) + 20*(194059800*x^4 + 1128781170*x^3 - 612106475*x^2 - 1785872944*x - 632498543)
*sqrt(5*x + 3)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*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)**5/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 2.00407, size = 265, normalized size = 1.87 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{199650000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{111321}{40000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (215622 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 205 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 741559591 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{665500000 \,{\left (2 \, x - 1\right )}} - \frac{337 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{16637500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{1011 \, \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}}}{12478125 \,{\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)^5/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/199650000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 111321/40000*sqrt(10)*arcsin(1/
11*sqrt(22)*sqrt(5*x + 3)) + 1/665500000*(215622*(12*sqrt(5)*(5*x + 3) + 205*sqrt(5))*(5*x + 3) - 741559591*sq
rt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 337/16637500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) + 1/12478125*(1011*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