3.2628 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{357 (3 x+2)^3}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{5281 \sqrt{1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (55300905 x+33035947)}{8784600 \sqrt{5 x+3}}+\frac{2997 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(5281*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*(3 + 5*x)^(3/2)) - (357*(2 + 3*x)^3)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(33035947 + 55300905*x))/(8784600*S
qrt[3 + 5*x]) + (2997*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.0407032, 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, 143, 54, 216} \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{357 (3 x+2)^3}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{5281 \sqrt{1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (55300905 x+33035947)}{8784600 \sqrt{5 x+3}}+\frac{2997 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5281*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*(3 + 5*x)^(3/2)) - (357*(2 + 3*x)^3)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(33035947 + 55300905*x))/(8784600*S
qrt[3 + 5*x]) + (2997*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*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 143

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x
)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] &&  !(SumSimplerQ[n, 1] &&  !SumSimplerQ[m, 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)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^3 \left (141+\frac{507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{357 (2+3 x)^3}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-6537-\frac{48861 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{5281 \sqrt{1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac{357 (2+3 x)^3}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\left (-\frac{1453983}{4}-\frac{5027355 x}{8}\right ) (2+3 x)}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{5281 \sqrt{1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac{357 (2+3 x)^3}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{\sqrt{1-2 x} (33035947+55300905 x)}{8784600 \sqrt{3+5 x}}+\frac{2997}{400} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{5281 \sqrt{1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac{357 (2+3 x)^3}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{\sqrt{1-2 x} (33035947+55300905 x)}{8784600 \sqrt{3+5 x}}+\frac{2997 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{200 \sqrt{5}}\\ &=\frac{5281 \sqrt{1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac{357 (2+3 x)^3}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{\sqrt{1-2 x} (33035947+55300905 x)}{8784600 \sqrt{3+5 x}}+\frac{2997 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{200 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 1.18252, size = 257, normalized size = 1.81 \[ -\frac{37 \left (-1320000 (3 x+2)^3 (1-2 x)^{7/2} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},2,2,\frac{7}{2}\right \},\left \{1,1,\frac{9}{2}\right \},\frac{5}{11} (1-2 x)\right )-1050000 (x+3) \left (6 x^2+x-2\right )^2 (1-2 x)^{5/2} \, _2F_1\left (\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{5}{11} (1-2 x)\right )+77 \sqrt{55} \left (\sqrt{10-20 x} \sqrt{5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )-27951 \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )\right )}{614922000 \sqrt{22} (1-2 x)^3}+\frac{1183 \left (19573 x^3+62232 x^2+52044 x+13040\right )}{878460 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{3 (3 x+2)^4}{10 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(2 + 3*x)^4)/(10*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (1183*(13040 + 52044*x + 62232*x^2 + 19573*x^3))/(8784
60*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (37*(77*Sqrt[55]*(Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*(1669914 + 2053496*x + 1
229303*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*Hypergeometric2F1[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]))/(614922000*Sqrt[22]*(1 - 2*x)^3)

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Maple [A]  time = 0.014, size = 182, normalized size = 1.3 \begin{align*}{\frac{1}{175692000\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 13163723100\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-4269315600\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-7766596629\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-789823386\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+25208605020\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -391879320\,x\sqrt{-10\,{x}^{2}-x+3}-3366379220\,\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)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/175692000*(1-2*x)^(1/2)*(13163723100*10^(1/2)*arcsin(20/11*x+1/11)*x^4+2632744620*10^(1/2)*arcsin(20/11*x+1/
11)*x^3-4269315600*x^4*(-10*x^2-x+3)^(1/2)-7766596629*10^(1/2)*arcsin(20/11*x+1/11)*x^2+24956232800*x^3*(-10*x
^2-x+3)^(1/2)-789823386*10^(1/2)*arcsin(20/11*x+1/11)*x+25208605020*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2
)*arcsin(20/11*x+1/11)-391879320*x*(-10*x^2-x+3)^(1/2)-3366379220*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)
^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.60381, size = 266, normalized size = 1.87 \begin{align*} -\frac{243 \, x^{4}}{10 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{999}{5856400} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{2997}{4000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{360639}{2928200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{5842159 \, x}{878460 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3429 \, x^{2}}{25 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{947293}{21961500 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3016649 \, x}{90750 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1851167}{90750 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

-243/10*x^4/(-10*x^2 - x + 3)^(3/2) + 999/5856400*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x +
3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 2997
/4000*sqrt(10)*arcsin(-20/11*x - 1/11) + 360639/2928200*sqrt(-10*x^2 - x + 3) - 5842159/878460*x/sqrt(-10*x^2
- x + 3) + 3429/25*x^2/(-10*x^2 - x + 3)^(3/2) + 947293/21961500/sqrt(-10*x^2 - x + 3) + 3016649/90750*x/(-10*
x^2 - x + 3)^(3/2) - 1851167/90750/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.76098, size = 404, normalized size = 2.85 \begin{align*} -\frac{131637231 \, \sqrt{10}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, 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 (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{175692000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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)^(5/2),x, algorithm="fricas")

[Out]

-1/175692000*(131637231*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*
x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(213465780*x^4 - 1247811640*x^3 - 1260430251*x^2 + 19593966*x + 1
68318961)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*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)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 2.43441, size = 265, normalized size = 1.87 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{439230000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{2997}{2000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{31 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{3327500 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (10673289 \, \sqrt{5}{\left (5 \, x + 3\right )} - 440040554 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7233942969 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5490375000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{1023 \, \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}}}{27451875 \,{\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)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/439230000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 2997/2000*sqrt(10)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3)) - 31/3327500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/549037500
0*(4*(10673289*sqrt(5)*(5*x + 3) - 440040554*sqrt(5))*(5*x + 3) + 7233942969*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x
+ 5)/(2*x - 1)^2 + 1/27451875*(1023*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