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

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

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Rubi [A]  time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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 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 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 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*}

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Mathematica [C]  time = 1.71, size = 275, normalized size = 1.94 \begin {gather*} \frac {\frac {173 \left (1320000 \sqrt {22} (2 x-1) (3 x+2)^3 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};-\frac {5}{11} (2 x-1)\right )-1050000 \sqrt {22} (x+3) \left (6 x^2+x-2\right )^2 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {5}{11} (2 x-1)\right )-\frac {23674497 \sqrt {10} \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{(1-2 x)^{5/2}}+\frac {8470 \sqrt {5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )}{(1-2 x)^2}\right )}{1089}+2079 (1-2 x) \left (\frac {10 \left (49005 x^2+60010 x+18373\right )}{(5 x+3)^{3/2}}+\frac {29403 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right )-\frac {7986000 (3 x+2)^4}{(5 x+3)^{3/2}}}{53240000 \sqrt {1-2 x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-7986000*(2 + 3*x)^4)/(3 + 5*x)^(3/2) + 2079*(1 - 2*x)*((10*(18373 + 60010*x + 49005*x^2))/(3 + 5*x)^(3/2) +
(29403*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/Sqrt[-1 + 2*x]) + (173*((8470*Sqrt[3 + 5*x]*(1669914 + 20
53496*x + 1229303*x^2 - 400032*x^3 + 28080*x^4 + 43200*x^5))/(1 - 2*x)^2 - (23674497*Sqrt[10]*(374 + 1296*x +
513*x^2 + 108*x^3)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1 - 2*x)^(5/2) - 1050000*Sqrt[22]*(3 + x)*(-2 + x + 6*x^
2)^2*Hypergeometric2F1[3/2, 9/2, 11/2, (-5*(-1 + 2*x))/11] + 1320000*Sqrt[22]*(-1 + 2*x)*(2 + 3*x)^3*Hypergeom
etricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (-5*(-1 + 2*x))/11]))/1089)/(53240000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.19, size = 141, normalized size = 0.99 \begin {gather*} \frac {\sqrt {5 x+3} \left (-\frac {1600 (1-2 x)^4}{(5 x+3)^4}-\frac {163520 (1-2 x)^3}{(5 x+3)^3}+\frac {2224548725 (1-2 x)^2}{(5 x+3)^2}+\frac {1487980494 (1-2 x)}{5 x+3}[phone]\right )}{15972000 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}+\frac {111321 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{4000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 5*x]*(201684000 - (1600*(1 - 2*x)^4)/(3 + 5*x)^4 - (163520*(1 - 2*x)^3)/(3 + 5*x)^3 + (2224548725*(1
- 2*x)^2)/(3 + 5*x)^2 + (1487980494*(1 - 2*x))/(3 + 5*x)))/(15972000*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*
x))^2) + (111321*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(4000*Sqrt[10])

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fricas [A]  time = 0.83, size = 111, normalized size = 0.78 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.50, size = 191, normalized size = 1.35 \begin {gather*} -\frac {1}{199650000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4044 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \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 {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1011 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{12478125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

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) + 4044*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3)) - 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*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1
/12478125*sqrt(10)*(5*x + 3)^(3/2)*(1011*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-
10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.02, size = 168, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (-3881196000 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-22575623400 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-5334057036 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x -4000542777 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{319440000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/319440000*(-2*x+1)^(1/2)*(22225237650*10^(1/2)*x^3*arcsin(20/11*x+1/11)-3881196000*(-10*x^2-x+3)^(1/2)*x^4+
15557666355*10^(1/2)*x^2*arcsin(20/11*x+1/11)-22575623400*(-10*x^2-x+3)^(1/2)*x^3-5334057036*10^(1/2)*x*arcsin
(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x^2-4000542777*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x
^2-x+3)^(1/2)*x+12649970860*(-10*x^2-x+3)^(1/2))/(2*x-1)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [A]  time = 1.36, size = 112, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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