3.27.27 \(\int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [2627]

Optimal. Leaf size=171 \[ \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}} \]

[Out]

7/33*(2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)^(3/2)+753543/80000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-511/242*(
2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2)+7591/39930*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(3/2)+261331/2196150*(2+3*x)^2
*(1-2*x)^(1/2)/(3+5*x)^(1/2)-7/117128000*(190406711+78981180*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 152, 56, 222} \begin {gather*} \frac {753543 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}}+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {511 (3 x+2)^4}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {7591 \sqrt {1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac {261331 \sqrt {1-2 x} (3 x+2)^2}{2196150 \sqrt {5 x+3}}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)}{117128000} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(7591*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(39930*(3 + 5*x)^(3/2)) - (511*(2 + 3*x)^4)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (261331*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(2196150*Sqrt[3
+ 5*x]) - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x))/117128000 + (753543*ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]])/(8000*Sqrt[10])

Rule 56

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 100

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 152

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 155

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 222

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)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^4 \left (204+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {32415}{2}-\frac {115641 x}{4}\right ) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-880026-\frac {11995011 x}{8}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {4 \int \frac {\left (-\frac {127241163}{8}-\frac {414651195 x}{16}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3294225}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000 \sqrt {5}}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 79, normalized size = 0.46 \begin {gather*} -\frac {44437106459+19932058554 x-274128335769 x^2-252342435560 x^3+97980793020 x^4+12807946800 x^5}{351384000 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {753543 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{8000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

-1/351384000*(44437106459 + 19932058554*x - 274128335769*x^2 - 252342435560*x^3 + 97980793020*x^4 + 1280794680
0*x^5)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (753543*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(8000*Sqrt[10])

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Maple [A]
time = 0.09, size = 199, normalized size = 1.16

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \left (3309786918900 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}-256158936000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-1959615860400 x^{4} \sqrt {-10 x^{2}-x +3}-1952774282151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}-198587215134 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-398641171080 x \sqrt {-10 x^{2}-x +3}-888742129180 \sqrt {-10 x^{2}-x +3}\right )}{7027680000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/7027680000*(1-2*x)^(1/2)*(3309786918900*10^(1/2)*arcsin(20/11*x+1/11)*x^4-256158936000*x^5*(-10*x^2-x+3)^(1/
2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^3-1959615860400*x^4*(-10*x^2-x+3)^(1/2)-1952774282151*10^(1/2)
*arcsin(20/11*x+1/11)*x^2+5046848711200*x^3*(-10*x^2-x+3)^(1/2)-198587215134*10^(1/2)*arcsin(20/11*x+1/11)*x+5
482566715380*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-398641171080*x*(-10*x^2-x+3)^(
1/2)-888742129180*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]
time = 0.50, size = 214, normalized size = 1.25 \begin {gather*} -\frac {729 \, x^{5}}{20 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {111537 \, x^{4}}{400 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {251181}{234256000} \, 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 {753543}{160000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {90676341}{117128000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {170985889 \, x}{7027680 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {766611 \, x^{2}}{1000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1005653687}{878460000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {416356591 \, x}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {496819753}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-729/20*x^5/(-10*x^2 - x + 3)^(3/2) - 111537/400*x^4/(-10*x^2 - x + 3)^(3/2) + 251181/234256000*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)) - 753543/160000*sqrt(10)*arcsin(-20/11*x - 1/11) + 90676341/117128000*
sqrt(-10*x^2 - x + 3) - 170985889/7027680*x/sqrt(-10*x^2 - x + 3) + 766611/1000*x^2/(-10*x^2 - x + 3)^(3/2) +
1005653687/878460000/sqrt(-10*x^2 - x + 3) + 416356591/3630000*x/(-10*x^2 - x + 3)^(3/2) - 496819753/3630000/(
-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.45, size = 126, normalized size = 0.74 \begin {gather*} -\frac {33097869189 \, \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 (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7027680000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/7027680000*(33097869189*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*(12807946800*x^5 + 97980793020*x^4 - 252342435560*x^3 - 274128
335769*x^2 + 19932058554*x + 44437106459)*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)] Timed out
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)**6/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 1.44, size = 204, normalized size = 1.19 \begin {gather*} -\frac {1}{2196150000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4884 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {753543}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (32019867 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 110347010662 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1221 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{137259375 \, {\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)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/2196150000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 4884*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3)) + 753543/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/219615000000*(4*(3
2019867*(4*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) - 110347010662*sqrt(5))*(5*x + 3) + 1820310410259*sqrt(5)
)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/137259375*sqrt(10)*(5*x + 3)^(3/2)*(1221*(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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^6}{{\left (1-2\,x\right )}^{5/2}\,{\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)^6/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)

[Out]

int((3*x + 2)^6/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)), x)

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