3.25.7 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\) [2407]

Optimal. Leaf size=178 \[ -\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac {40}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {3244595 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{108864 \sqrt {7}} \]

[Out]

-1/12*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4+115/216*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-3244595/762048*arcta
n(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-40/243*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+2675/86
4*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-97235/36288*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.04, antiderivative size = 178, 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, 154, 163, 56, 222, 95, 210} \begin {gather*} -\frac {40}{243} \sqrt {10} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {3244595 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{108864 \sqrt {7}}-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac {2675 (5 x+3)^{3/2} \sqrt {1-2 x}}{864 (3 x+2)^2}-\frac {97235 \sqrt {5 x+3} \sqrt {1-2 x}}{36288 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-97235*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(12*(2 + 3*x)^4) +
(115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(216*(2 + 3*x)^3) + (2675*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(864*(2 + 3*x)^
2) - (40*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (3244595*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(108864*Sqrt[7])

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 154

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] && ILtQ[m, -1] && GtQ[n, 0]

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)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}-\frac {1}{108} \int \frac {\left (-\frac {3315}{4}-240 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac {1}{648} \int \frac {\left (\frac {92115}{8}-960 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac {\int \frac {\frac {2886195}{16}-33600 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{13608}\\ &=-\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac {200}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {3244595 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{217728}\\ &=-\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac {3244595 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{108864}-\frac {1}{243} \left (80 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac {40}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {3244595 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{108864 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 108, normalized size = 0.61 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (677168+2947548 x+4103592 x^2+1790325 x^3\right )}{(2+3 x)^4}+125440 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-3244595 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{762048} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(677168 + 2947548*x + 4103592*x^2 + 1790325*x^3))/(2 + 3*x)^4 + 125440*Sqrt[1
0]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 3244595*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/7620
48

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(136)=272\).
time = 0.12, size = 315, normalized size = 1.77

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (1790325 x^{3}+4103592 x^{2}+2947548 x +677168\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{36288 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {20 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}-\frac {3244595 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{1524096}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(143\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (262812195 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-10160640 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-27095040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-27095040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -12042240 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2007040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{1524096 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1524096*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(262812195*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4-10160640*10^(1/2)*arcsin(20/11*x+1/11)*x^4+700832520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^3-27095040*10^(1/2)*arcsin(20/11*x+1/11)*x^3+700832520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^2-27095040*10^(1/2)*arcsin(20/11*x+1/11)*x^2+75193650*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-12042240*10^(1/2)*arcsin(20/11*x+1/11)*x+172350864*x^2*(-10
*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-2007040*10^(1/2)*arcsin(20
/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]
time = 0.56, size = 197, normalized size = 1.11 \begin {gather*} \frac {21775}{21168} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {95 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{168 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4355 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4704 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {539675}{42336} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {20}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3244595}{1524096} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1460395}{254016} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {18245 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28224 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21775/21168*(-10*x^2 - x + 3)^(3/2) + 1/4*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 9
5/168*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4355/4704*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x +
4) + 539675/42336*sqrt(-10*x^2 - x + 3)*x - 20/243*sqrt(10)*arcsin(20/11*x + 1/11) + 3244595/1524096*sqrt(7)*
arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1460395/254016*sqrt(-10*x^2 - x + 3) + 18245/28224*(-10*x^
2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]
time = 0.63, size = 176, normalized size = 0.99 \begin {gather*} -\frac {3244595 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 ) - 125440 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 ) - 42 \, {\left (1790325 \, x^{3} + 4103592 \, x^{2} + 2947548 \, x + 677168\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1524096 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1524096*(3244595*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 125440*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*s
qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(1790325*x^3 + 4103592*x^2 + 2947548*x
+ 677168)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (136) = 272\).
time = 1.20, size = 435, normalized size = 2.44 \begin {gather*} \frac {648919}{3048192} \, \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 {20}{243} \, \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 {55 \, \sqrt {10} {\left (19447 \, {\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 )}^{7} + 19946472 \, {\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 )}^{5} - 6199166400 \, {\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 )}^{3} - \frac {348224576000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1392898304000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{18144 \, {\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 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

648919/3048192*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)))) - 20/243*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)))) - 55/18144*s
qrt(10)*(19447*((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)))^7 + 19946472*((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)))^5 - 6199166400*((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)))^3 - 348224576000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 139
2898304000*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)^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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