3.24.54 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\) [2354]

Optimal. Leaf size=159 \[ -\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {43 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]

[Out]

-1/3*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)-43/7776*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-181/243*arctan(1
/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+25/12*(3+5*x)^(3/2)*(1-2*x)^(1/2)-8/27*(3+5*x)^(5/2)*(1-2*x)^(
1/2)-3065/1296*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 159, 163, 56, 222, 95, 210} \begin {gather*} -\frac {43 \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3888}-\frac {181}{243} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {8}{27} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {25}{12} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {3065 \sqrt {1-2 x} \sqrt {5 x+3}}{1296} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3065*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1296 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12 - (8*Sqrt[1 - 2*x]*(3 + 5*x)^
(5/2))/27 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) - (43*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
3888 - (181*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 99

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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{135} \int \frac {\left (\frac {1835}{2}-3375 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {\int \frac {\left (-2655-\frac {45975 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1620}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {\int \frac {\frac {49605}{2}-\frac {3225 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{9720}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {215 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{7776}+\frac {1267}{486} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1267}{243} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {\left (43 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3888}\\ &=-\frac {3065 \sqrt {1-2 x} \sqrt {3+5 x}}{1296}+\frac {25}{12} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {43 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3888}-\frac {181}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 113, normalized size = 0.71 \begin {gather*} \frac {-\frac {6 \sqrt {1-2 x} \left (2190-6889 x-23145 x^2+12300 x^3+36000 x^4\right )}{(2+3 x) \sqrt {3+5 x}}+43 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-5792 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7776} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[1 - 2*x]*(2190 - 6889*x - 23145*x^2 + 12300*x^3 + 36000*x^4))/((2 + 3*x)*Sqrt[3 + 5*x]) + 43*Sqrt[10
]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 5792*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/7776

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Maple [A]
time = 0.14, size = 180, normalized size = 1.13

method	result	size
risch	\(\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (7200 x^{3}-1860 x^{2}-3513 x +730\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1296 \left (2+3 x \right ) \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (-\frac {43 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{15552}+\frac {181 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{486}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(142\)
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (86400 x^{3} \sqrt {-10 x^{2}-x +3}+129 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -17376 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -22320 x^{2} \sqrt {-10 x^{2}-x +3}+86 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11584 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-42156 x \sqrt {-10 x^{2}-x +3}+8760 \sqrt {-10 x^{2}-x +3}\right )}{15552 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15552*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(86400*x^3*(-10*x^2-x+3)^(1/2)+129*10^(1/2)*arcsin(20/11*x+1/11)*x-17376*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-22320*x^2*(-10*x^2-x+3)^(1/2)+86*10^(1/2)*arcsin(
20/11*x+1/11)-11584*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-42156*x*(-10*x^2-x+3)^(1/2)+876
0*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]
time = 0.50, size = 104, normalized size = 0.65 \begin {gather*} \frac {5}{27} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {245}{108} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {43}{15552} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {181}{486} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1301}{1296} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/27*(-10*x^2 - x + 3)^(3/2) + 245/108*sqrt(-10*x^2 - x + 3)*x - 43/15552*sqrt(10)*arcsin(20/11*x + 1/11) + 18
1/486*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1301/1296*sqrt(-10*x^2 - x + 3) + 1/9*(-10*x
^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]
time = 0.73, size = 137, normalized size = 0.86 \begin {gather*} \frac {43 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5792 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 12 \, {\left (7200 \, x^{3} - 1860 \, x^{2} - 3513 \, x + 730\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15552 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15552*(43*sqrt(5)*sqrt(2)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*
x^2 + x - 3)) - 5792*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 12*(7200*x^3 - 1860*x^2 - 3513*x + 730)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (115) = 230\).
time = 1.33, size = 305, normalized size = 1.92 \begin {gather*} \frac {181}{4860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{2160} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 85 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 835 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {43}{15552} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {154 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{81 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

181/4860*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^
2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1/2160*(4*(8*sqrt(5)*(5*x + 3) - 85*sqrt(5))*(5*x +
3) + 835*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 43/15552*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2
)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 154/81*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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