3.2618 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1561 (3 x+2)^3}{726 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{7723 \sqrt{1-2 x} (3 x+2)^2}{39930 \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (16227780 x+39109961)}{2129600}+\frac{243189 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

[Out]

(7723*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*Sqrt[3 + 5*x]) - (1561*(2 + 3*x)^3)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
+ (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39109961 + 16227780*x))/2
129600 + (243189*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi [A]  time = 0.0425497, 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} \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1561 (3 x+2)^3}{726 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{7723 \sqrt{1-2 x} (3 x+2)^2}{39930 \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (16227780 x+39109961)}{2129600}+\frac{243189 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7723*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*Sqrt[3 + 5*x]) - (1561*(2 + 3*x)^3)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
+ (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39109961 + 16227780*x))/2
129600 + (243189*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*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)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^3 \left (211+\frac{717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{1561 (2+3 x)^3}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-17212-\frac{117321 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{7723 \sqrt{1-2 x} (2+3 x)^2}{39930 \sqrt{3+5 x}}-\frac{1561 (2+3 x)^3}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2 \int \frac{\left (-\frac{1244193}{4}-\frac{4056945 x}{8}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=\frac{7723 \sqrt{1-2 x} (2+3 x)^2}{39930 \sqrt{3+5 x}}-\frac{1561 (2+3 x)^3}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (39109961+16227780 x)}{2129600}+\frac{243189 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3200}\\ &=\frac{7723 \sqrt{1-2 x} (2+3 x)^2}{39930 \sqrt{3+5 x}}-\frac{1561 (2+3 x)^3}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (39109961+16227780 x)}{2129600}+\frac{243189 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1600 \sqrt{5}}\\ &=\frac{7723 \sqrt{1-2 x} (2+3 x)^2}{39930 \sqrt{3+5 x}}-\frac{1561 (2+3 x)^3}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (39109961+16227780 x)}{2129600}+\frac{243189 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1600 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 3.47368, size = 164, normalized size = 1.15 \[ \frac{83650000 \sqrt{22} (1-2 x)^{9/2} (3 x+2)^3 (5 x+3) \, _2F_1\left (\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{5}{11} (1-2 x)\right )-363 \left (10 \sqrt{1-2 x} \sqrt{5 x+3} \left (180684000 x^6-102226860 x^5+1439225865 x^4+557838977 x^3+7613332652 x^2+5247676119 x+325823031\right )+3993 \sqrt{10} \left (1822635 x^4-29307609 x^3-21484944 x^2-5504683 x-2134887\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{23191344000 (2 x-1)^3 (5 x+3)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-363*(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(325823031 + 5247676119*x + 7613332652*x^2 + 557838977*x^3 + 1439225865*
x^4 - 102226860*x^5 + 180684000*x^6) + 3993*Sqrt[10]*(-2134887 - 5504683*x - 21484944*x^2 - 29307609*x^3 + 182
2635*x^4)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]) + 83650000*Sqrt[22]*(1 - 2*x)^(9/2)*(2 + 3*x)^3*(3 + 5*x)*Hypergeo
metric2F1[3/2, 9/2, 11/2, (5*(1 - 2*x))/11])/(23191344000*(-1 + 2*x)^3*(3 + 5*x))

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Maple [A]  time = 0.014, size = 168, normalized size = 1.2 \begin{align*}{\frac{1}{127776000\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 19421073540\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-1552478400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-7768429416\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-10737975600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-6797375739\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+35819748080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-8705162580\,\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)^5/(1-2*x)^(5/2)/(3+5*x)^(3/2),x)

[Out]

1/127776000*(1-2*x)^(1/2)*(19421073540*10^(1/2)*arcsin(20/11*x+1/11)*x^3-1552478400*x^4*(-10*x^2-x+3)^(1/2)-77
68429416*10^(1/2)*arcsin(20/11*x+1/11)*x^2-10737975600*x^3*(-10*x^2-x+3)^(1/2)-6797375739*10^(1/2)*arcsin(20/1
1*x+1/11)*x+35819748080*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^
2-x+3)^(1/2)-8705162580*(-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 = 4.63498, size = 151, normalized size = 1.06 \begin{align*} \frac{243 \, x^{3}}{40 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{243189}{32000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{7209 \, x^{2}}{160 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{751566017 \, x}{6388800 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{638622829}{6388800 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16807}{528 \,{\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)^5/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

243/40*x^3/sqrt(-10*x^2 - x + 3) + 243189/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 7209/160*x^2/sqrt(-10
*x^2 - x + 3) - 751566017/6388800*x/sqrt(-10*x^2 - x + 3) - 638622829/6388800/sqrt(-10*x^2 - x + 3) - 16807/52
8/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.53345, size = 373, normalized size = 2.63 \begin{align*} -\frac{971053677 \, \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 (77623920 \, x^{4} + 536898780 \, x^{3} - 1790987404 \, x^{2} - 525679641 \, x + 435258129\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{127776000 \,{\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)^5/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/127776000*(971053677*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*(77623920*x^4 + 536898780*x^3 - 1790987404*x^2 - 525679641*x + 435258129)*sq
rt(5*x + 3)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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

[Out]

Timed out

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Giac [A]  time = 2.10701, size = 194, normalized size = 1.37 \begin{align*} \frac{243189}{16000} \, \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 )}}{1663750 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (323433 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 271 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 3237172310 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 53407238379 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3993000000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{831875 \,{\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)^5/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

243189/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1663750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3) - 1/3993000000*(4*(323433*(12*sqrt(5)*(5*x + 3) + 271*sqrt(5))*(5*x + 3) - 3237172310*sqrt
(5))*(5*x + 3) + 53407238379*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/831875*sqrt(10)*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))