3.25.6 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\) [2406]

Optimal. Leaf size=171 \[ -\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {362}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {215 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1944 \sqrt {7}} \]

[Out]

-1/9*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^3+115/108*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^2+215/13608*arctan(1/7*
(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+362/243*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+365/216*(3+5
*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)-845/648*(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 = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {99, 154, 159, 163, 56, 222, 95, 210} \begin {gather*} \frac {362}{243} \sqrt {10} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {215 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1944 \sqrt {7}}-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac {365 (5 x+3)^{3/2} \sqrt {1-2 x}}{216 (3 x+2)}-\frac {845}{648} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/648 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (115*(1 - 2*x)^(3
/2)*(3 + 5*x)^(3/2))/(108*(2 + 3*x)^2) + (365*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(216*(2 + 3*x)) + (362*Sqrt[10]*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 + (215*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1944*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 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)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {2325}{4}-735 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {1}{162} \int \frac {\sqrt {3+5 x} \left (\frac {6975}{8}+\frac {2535 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {1}{972} \int \frac {-\frac {57705}{4}-21720 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {215 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{3888}+\frac {1810}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {215 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1944}+\frac {1}{243} \left (724 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {362}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {215 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1944 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 108, normalized size = 0.63 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (10304+36234 x+34341 x^2+4320 x^3\right )}{(2+3 x)^3}-20272 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+215 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(10304 + 36234*x + 34341*x^2 + 4320*x^3))/(2 + 3*x)^3 - 20272*Sqrt[10]*ArcTan
[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 215*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/13608

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(269\) vs. \(2(129)=258\).
time = 0.12, size = 270, normalized size = 1.58

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (4320 x^{3}+34341 x^{2}+36234 x +10304\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{648 \left (2+3 x \right )^{3} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {181 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}+\frac {215 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{27216}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(143\)
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5805 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-547344 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+11610 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-1094688 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-181440 x^{3} \sqrt {-10 x^{2}-x +3}+7740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -729792 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1442322 x^{2} \sqrt {-10 x^{2}-x +3}+1720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-162176 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1521828 x \sqrt {-10 x^{2}-x +3}-432768 \sqrt {-10 x^{2}-x +3}\right )}{27216 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\)	\(270\)

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)^4,x,method=_RETURNVERBOSE)

[Out]

-1/27216*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5805*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-5473
44*10^(1/2)*arcsin(20/11*x+1/11)*x^3+11610*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1094
688*10^(1/2)*arcsin(20/11*x+1/11)*x^2-181440*x^3*(-10*x^2-x+3)^(1/2)+7740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x-729792*10^(1/2)*arcsin(20/11*x+1/11)*x-1442322*x^2*(-10*x^2-x+3)^(1/2)+1720*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-162176*10^(1/2)*arcsin(20/11*x+1/11)-1521828*x*(-10*x^2-x+3)
^(1/2)-432768*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]
time = 0.57, size = 161, normalized size = 0.94 \begin {gather*} \frac {125}{378} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{84 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1825}{756} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {181}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {215}{27216} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {655}{4536} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {65 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{504 \, {\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)^4,x, algorithm="maxima")

[Out]

125/378*(-10*x^2 - x + 3)^(3/2) + 1/3*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 25/84*(-10*x^2 -
x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 1825/756*sqrt(-10*x^2 - x + 3)*x + 181/243*sqrt(10)*arcsin(20/11*x + 1/11) -
215/27216*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 655/4536*sqrt(-10*x^2 - x + 3) - 65/504
*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]
time = 0.70, size = 161, normalized size = 0.94 \begin {gather*} \frac {215 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 ) - 20272 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (4320 \, x^{3} + 34341 \, x^{2} + 36234 \, x + 10304\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{27216 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

1/27216*(215*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) - 20272*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)
*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(4320*x^3 + 34341*x^2 + 36234*x + 10304)*sqrt(5*x + 3)*sqrt(-2*x + 1))/
(27*x^3 + 54*x^2 + 36*x + 8)

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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)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (129) = 258\).
time = 1.71, size = 396, normalized size = 2.32 \begin {gather*} -\frac {43}{54432} \, \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 {181}{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 {4}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (67 \, {\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} + 56000 \, {\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 {65464000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {261856000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{108 \, {\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 )}^{3}} \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)^4,x, algorithm="giac")

[Out]

-43/54432*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)))) + 181/243*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)))) + 4/81*sqrt(5)*s
qrt(5*x + 3)*sqrt(-10*x + 5) - 11/108*sqrt(10)*(67*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 56000*((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 - 65464000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) + 261856000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3

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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 )}^4} \,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)^4,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^4, x)

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