3.27.6 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\) [2606]

Optimal. Leaf size=173 \[ \frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {2585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}} \]

[Out]

11/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^3-2585/134456*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2
)+15755/86436*(3+5*x)^(1/2)/(1-2*x)^(1/2)+32/441*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)-187/588*(3+5*x)^(1/2)/(
2+3*x)^2/(1-2*x)^(1/2)-2365/8232*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 154, 156, 157, 12, 95, 210} \begin {gather*} -\frac {2585 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}}+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac {15755 \sqrt {5 x+3}}{86436 \sqrt {1-2 x}}-\frac {2365 \sqrt {5 x+3}}{8232 \sqrt {1-2 x} (3 x+2)}-\frac {187 \sqrt {5 x+3}}{588 \sqrt {1-2 x} (3 x+2)^2}+\frac {32 \sqrt {5 x+3}}{441 \sqrt {1-2 x} (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15755*Sqrt[3 + 5*x])/(86436*Sqrt[1 - 2*x]) + (32*Sqrt[3 + 5*x])/(441*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (187*Sqrt[3
+ 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (2365*Sqrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)
^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (2585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7]
)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 156

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1}{21} \int \frac {\left (-123-\frac {465 x}{2}\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {24783}{2}-\frac {43065 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1323}\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {264495}{4}-117810 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{18522}\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {2149455}{8}-\frac {744975 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{129654}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {\int \frac {5374215}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4991679}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2585 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38416}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2585 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{19208}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {2585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 2.51, size = 152, normalized size = 0.88 \begin {gather*} \frac {5 \left (-\frac {7 \sqrt {3+5 x} \left (-75888-304730 x-169221 x^2+552780 x^3+567180 x^4\right )}{5 (1-2 x)^{3/2} (2+3 x)^3}+1551 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+1551 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{403368} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*((-7*Sqrt[3 + 5*x]*(-75888 - 304730*x - 169221*x^2 + 552780*x^3 + 567180*x^4))/(5*(1 - 2*x)^(3/2)*(2 + 3*x)
^3) + 1551*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 1551*Sqrt[
7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/403368

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(134)=268\).
time = 0.08, size = 305, normalized size = 1.76

method	result	size
default	\(\frac {\left (837540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+837540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-348975 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-7940520 x^{4} \sqrt {-10 x^{2}-x +3}-449790 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-7738920 x^{3} \sqrt {-10 x^{2}-x +3}+31020 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2369094 x^{2} \sqrt {-10 x^{2}-x +3}+62040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4266220 x \sqrt {-10 x^{2}-x +3}+1062432 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{806736 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/806736*(837540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+837540*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-348975*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-
7940520*x^4*(-10*x^2-x+3)^(1/2)-449790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-7738920*
x^3*(-10*x^2-x+3)^(1/2)+31020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2369094*x^2*(-10*x^
2-x+3)^(1/2)+62040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4266220*x*(-10*x^2-x+3)^(1/2)+10
62432*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.52, size = 240, normalized size = 1.39 \begin {gather*} \frac {2585}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {78775 \, x}{86436 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {11755}{172872 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {17875 \, x}{12348 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{1701 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {239}{15876 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {4997}{31752 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {901885}{666792 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

2585/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 78775/86436*x/sqrt(-10*x^2 - x + 3) +
11755/172872/sqrt(-10*x^2 - x + 3) + 17875/12348*x/(-10*x^2 - x + 3)^(3/2) - 1/1701/(27*(-10*x^2 - x + 3)^(3/2
)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 239/15876
/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) - 4997/31752/(3*(-
10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 901885/666792/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.48, size = 131, normalized size = 0.76 \begin {gather*} -\frac {7755 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, 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 ) + 14 \, {\left (567180 \, x^{4} + 552780 \, x^{3} - 169221 \, x^{2} - 304730 \, x - 75888\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{806736 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/806736*(7755*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5
*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(567180*x^4 + 552780*x^3 - 169221*x^2 - 304730*x - 75888)*sqrt(5
*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(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. 349 vs. \(2 (134) = 268\).
time = 3.69, size = 349, normalized size = 2.02 \begin {gather*} \frac {517}{537824} \, \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 {88 \, {\left (151 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1023 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1260525 \, {\left (2 \, x - 1\right )}^{2}} - \frac {11 \, \sqrt {10} {\left (3629 \, {\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} + 2900800 \, {\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 {755384000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3021536000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\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((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

517/537824*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)))) - 88/1260525*(151*sqrt(5)*(5*x + 3) - 1023*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 11/67228*sqrt(10)*(3629*((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 + 2900800*((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 + 755384000*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 3021536000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/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 (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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