3.2599 \(\int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=157 \[ \frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {9444023 \sqrt {1-2 x} (5 x+3)^{3/2}}{33792}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac {9444023 \sqrt {1-2 x} \sqrt {5 x+3}}{4096}+\frac {103884253 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4096 \sqrt {10}} \]

[Out]

1/3*(2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(3/2)[phone]/40960*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-373/66*
(2+3*x)^2*(3+5*x)^(5/2)/(1-2*x)^(1/2)-9444023/33792*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1/1408*(3+5*x)^(5/2)*(81191+40
164*x)*(1-2*x)^(1/2)-9444023/4096*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 150, 147, 50, 54, 216} \[ \frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {9444023 \sqrt {1-2 x} (5 x+3)^{3/2}}{33792}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac {9444023 \sqrt {1-2 x} \sqrt {5 x+3}}{4096}+\frac {103884253 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4096 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9444023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096 - (9444023*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/33792 - (373*(2 + 3*x)^2
*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 +
5*x)^(5/2)*(81191 + 40164*x))/1408 + (103884253*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096*Sqrt[10])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^2 (3+5 x)^{3/2} \left (52+\frac {165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {15989}{2}-\frac {50205 x}{4}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {9444023 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{8448}\\ &=-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {9444023 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{2048}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8192}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{4096 \sqrt {5}}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4096 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 100, normalized size = 0.64 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (1036800 x^5+5477760 x^4+15301008 x^3+40614996 x^2-129940960 x+47216961\right )[phone] \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{122880 \sqrt {1-2 x} (2 x-1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(47216961 - 129940960*x + 40614996*x^2 + 15301008*x^3 + 5477760*x^4 + 1036800
*x^5) + 311652759*Sqrt[10]*(1 - 2*x)^2*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(122880*Sqrt[1 - 2*x]*(-1 + 2*x)^(3
/2))

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fricas [A]  time = 1.20, size = 106, normalized size = 0.68 \[ -\frac {311652759 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (1036800 \, x^{5} + 5477760 \, x^{4} + 15301008 \, x^{3} + 40614996 \, x^{2} - 129940960 \, x + 47216961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{245760 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/245760*(311652759*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(
10*x^2 + x - 3)) + 20*(1036800*x^5 + 5477760*x^4 + 15301008*x^3 + 40614996*x^2 - 129940960*x + 47216961)*sqrt(
5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.19, size = 110, normalized size = 0.70 \[ \frac {103884253}{40960} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 137 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 13627 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9444023 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1038842530 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 17140901745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{7680000 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

103884253/40960*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/7680000*(4*(3*(36*(8*(12*sqrt(5)*(5*x + 3) +
137*sqrt(5))*(5*x + 3) + 13627*sqrt(5))*(5*x + 3) + 9444023*sqrt(5))*(5*x + 3) - 1038842530*sqrt(5))*(5*x + 3)
+ 17140901745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.02, size = 171, normalized size = 1.09 \[ \frac {\left (-20736000 \sqrt {-10 x^{2}-x +3}\, x^{5}-109555200 \sqrt {-10 x^{2}-x +3}\, x^{4}-306020160 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-812299920 \sqrt {-10 x^{2}-x +3}\, x^{2}-1246611036 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-944339220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{245760 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/245760*(-20736000*(-10*x^2-x+3)^(1/2)*x^5-109555200*(-10*x^2-x+3)^(1/2)*x^4+1246611036*10^(1/2)*x^2*arcsin(2
0/11*x+1/11)-306020160*(-10*x^2-x+3)^(1/2)*x^3-1246611036*10^(1/2)*x*arcsin(20/11*x+1/11)-812299920*(-10*x^2-x
+3)^(1/2)*x^2+311652759*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-944339220*(-10*x^2-x+3)
^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [C]  time = 1.68, size = 325, normalized size = 2.07 \[ \frac {2606989}{2048} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {395307}{81920} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {495}{256} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {63 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{64 \, {\left (2 \, x - 1\right )}} - \frac {16335}{1024} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {68607}{4096} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {114345}{512} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {18865 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{192 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {24255 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3465 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (2 \, x - 1\right )}} + \frac {207515 \, \sqrt {-10 \, x^{2} - x + 3}}{384 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3721795 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2606989/2048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 395307/81920*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) +
495/256*(-10*x^2 - x + 3)^(3/2) - 343/16*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 441/3
2*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 63/16*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 1) - 27/64
*(-10*x^2 - x + 3)^(5/2)/(2*x - 1) - 16335/1024*sqrt(10*x^2 - 21*x + 8)*x + 68607/4096*sqrt(10*x^2 - 21*x + 8)
- 114345/512*sqrt(-10*x^2 - x + 3) - 18865/192*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 24255/128
*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 3465/128*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 207515/384*sqrt(-10*
x^2 - x + 3)/(4*x^2 - 4*x + 1) + 3721795/768*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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