3.25.21 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\) [2421]

Optimal. Leaf size=209 \[ -\frac {805255 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}-\frac {8857805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]

[Out]

1/6*(1-2*x)^(5/2)*(3+5*x)^(7/2)/(2+3*x)^6+11/12*(1-2*x)^(3/2)*(3+5*x)^(7/2)/(2+3*x)^5-8857805/1229312*arctan(1
/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-73205/37632*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-1331/1344*(3
+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+121/32*(3+5*x)^(7/2)*(1-2*x)^(1/2)/(2+3*x)^4-805255/175616*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.05, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210} \begin {gather*} -\frac {8857805 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}}+\frac {121 \sqrt {1-2 x} (5 x+3)^{7/2}}{32 (3 x+2)^4}+\frac {11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{5/2}}{1344 (3 x+2)^3}-\frac {73205 \sqrt {1-2 x} (5 x+3)^{3/2}}{37632 (3 x+2)^2}-\frac {805255 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

(-805255*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (73205*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(37632*(2 + 3
*x)^2) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1344*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3
*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (121*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(32*(2 +
3*x)^4) - (8857805*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

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 96

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

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])

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {55}{12} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121}{8} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac {1331}{64} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac {73205 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{2688}\\ &=-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac {805255 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac {805255 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac {8857805 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {805255 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac {8857805 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {805255 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {73205 \sqrt {1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac {121 \sqrt {1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}-\frac {8857805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 89, normalized size = 0.43 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (79536960+589734736 x+1743189856 x^2+2573967504 x^3+1905431420 x^4+568572155 x^5\right )}{(2+3 x)^6}-26573415 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3687936} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(79536960 + 589734736*x + 1743189856*x^2 + 2573967504*x^3 + 1905431420*x^4 + 5
68572155*x^5))/(2 + 3*x)^6 - 26573415*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3687936

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(164)=328\).
time = 0.11, size = 346, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (568572155 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {8857805 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (19372019535 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(19372019535*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*
x^6+77488078140*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+129146796900*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+7960010170*x^5*(-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))*x^3+26676039880*x^4*(-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))*x^2+36035545056*x^3*(-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))*x+24404657984*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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]
time = 0.56, size = 302, normalized size = 1.44 \begin {gather*} \frac {3304795}{19361664} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{14 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {4387 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{10976 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {81733 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{153664 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {660959 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{4302592 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {59208325}{12907776} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {113659535}{25815552} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {109715471 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{77446656 \, {\left (3 \, x + 2\right )}} + \frac {13542925}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {8857805}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {11932415}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

3304795/19361664*(-10*x^2 - x + 3)^(5/2) + 1/14*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64) + 37/196*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32) + 4387/10976*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 81733/153664*(-10*x^2 -
x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 660959/4302592*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 592083
25/12907776*(-10*x^2 - x + 3)^(3/2)*x + 113659535/25815552*(-10*x^2 - x + 3)^(3/2) - 109715471/77446656*(-10*x
^2 - x + 3)^(5/2)/(3*x + 2) + 13542925/614656*sqrt(-10*x^2 - x + 3)*x + 8857805/2458624*sqrt(7)*arcsin(37/11*x
/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 11932415/1229312*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.50, size = 146, normalized size = 0.70 \begin {gather*} -\frac {26573415 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 ) - 14 \, {\left (568572155 \, x^{5} + 1905431420 \, x^{4} + 2573967504 \, x^{3} + 1743189856 \, x^{2} + 589734736 \, x + 79536960\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/7375872*(26573415*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*sq
rt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(568572155*x^5 + 1905431420*x^4 + 257396
7504*x^3 + 1743189856*x^2 + 589734736*x + 79536960)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x
^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**(5/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
time = 1.05, size = 484, normalized size = 2.32 \begin {gather*} \frac {1771561}{4917248} \, \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 {8857805 \, \sqrt {10} {\left (3 \, {\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 )}^{11} + 4760 \, {\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 )}^{9} + 3104640 \, {\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} - 869299200 \, {\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} - 104491520000 \, {\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 {5163110400000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {20652441600000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\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 )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

1771561/4917248*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 8857805/263424*sqrt(10)*(3*((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)))^11 + 4760*((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)))^9 + 3104640*
((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
- 869299200*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))^5 - 104491520000*((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 - 5163110400000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 20652441600000
*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)^6

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^7,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^7, x)

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