3.25.58 \(\int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac {465 \sqrt {5 x+3}}{9604 \sqrt {1-2 x}}-\frac {85 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {23 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {32 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]

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Rubi [A]  time = 0.06, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \begin {gather*} \frac {465 \sqrt {5 x+3}}{9604 \sqrt {1-2 x}}-\frac {85 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {23 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {32 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(465*Sqrt[3 + 5*x])/(9604*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (32*Sqrt[3 +
5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (23*Sqrt[3 + 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (85*Sqrt[3 + 5*x]
)/(2744*Sqrt[1 - 2*x]*(2 + 3*x)) - (9395*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 93

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 98

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 151

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] && IntegerQ[m]

Rule 152

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 204

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

Rubi steps

\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1}{21} \int \frac {-233-\frac {795 x}{2}}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{441} \int \frac {-\frac {3357}{2}-2880 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {\int \frac {-\frac {36855}{4}-14490 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{6174}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {268695}{8}-\frac {26775 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{43218}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {6510735}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1663893}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {9395 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38416}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {9395 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{19208}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {9395 \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 = 0.07, size = 100, normalized size = 0.58 \begin {gather*} -\frac {7 \sqrt {5 x+3} \left (150660 x^4+193860 x^3-17127 x^2-80510 x-19296\right )-28185 \sqrt {7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{403368 (1-2 x)^{3/2} (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/403368*(7*Sqrt[3 + 5*x]*(-19296 - 80510*x - 17127*x^2 + 193860*x^3 + 150660*x^4) - 28185*Sqrt[7 - 14*x]*(-1
+ 2*x)*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2*x)^(3/2)*(2 + 3*x)^3)

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IntegrateAlgebraic [A]  time = 0.28, size = 138, normalized size = 0.80 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {28185 (1-2 x)^4}{(5 x+3)^4}-\frac {526120 (1-2 x)^3}{(5 x+3)^3}-\frac {157143 (1-2 x)^2}{(5 x+3)^2}+\frac {931392 (1-2 x)}{5 x+3}+43904\right )}{57624 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(43904 - (28185*(1 - 2*x)^4)/(3 + 5*x)^4 - (526120*(1 - 2*x)^3)/(3 + 5*x)^3 - (157143*(1 - 2*
x)^2)/(3 + 5*x)^2 + (931392*(1 - 2*x))/(3 + 5*x)))/(57624*(1 - 2*x)^(3/2)*(7 + (1 - 2*x)/(3 + 5*x))^3) - (9395
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

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fricas [A]  time = 1.29, size = 131, normalized size = 0.76 \begin {gather*} -\frac {28185 \, \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 (150660 \, x^{4} + 193860 \, x^{3} - 17127 \, x^{2} - 80510 \, x - 19296\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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/806736*(28185*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*(150660*x^4 + 193860*x^3 - 17127*x^2 - 80510*x - 19296)*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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giac [B]  time = 3.25, size = 349, normalized size = 2.02 \begin {gather*} \frac {1879}{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 {8 \, {\left (512 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3201 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1260525 \, {\left (2 \, x - 1\right )}^{2}} - \frac {99 \, \sqrt {10} {\left (727 \, {\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} + 548800 \, {\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 {20776000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {83104000 \, \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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

1879/537824*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)))) - 8/1260525*(512*sqrt(5)*(5*x + 3) - 3201*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 99/67228*sqrt(10)*(727*((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 + 548800*((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)))^3 + 20776000*(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22))/sqrt(5*x + 3) - 83104000*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)^3

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maple [B]  time = 0.02, size = 305, normalized size = 1.76 \begin {gather*} \frac {\left (3043980 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3043980 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2109240 \sqrt {-10 x^{2}-x +3}\, x^{4}-1268325 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2714040 \sqrt {-10 x^{2}-x +3}\, x^{3}-1634730 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+239778 \sqrt {-10 x^{2}-x +3}\, x^{2}+112740 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1127140 \sqrt {-10 x^{2}-x +3}\, x +225480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+270144 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{806736 \left (3 x +2\right )^{3} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/806736*(3043980*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3043980*7^(1/2)*x^4*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1268325*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))-2109240*(-10*x^2-x+3)^(1/2)*x^4-1634730*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-2714
040*(-10*x^2-x+3)^(1/2)*x^3+112740*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+239778*(-10*x^
2-x+3)^(1/2)*x^2+225480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1127140*(-10*x^2-x+3)^(1/2)
*x+270144*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^3/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.53, size = 240, normalized size = 1.39 \begin {gather*} \frac {9395}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2325 \, x}{9604 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5395}{57624 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7625 \, x}{12348 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{567 \, {\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 {169}{5292 \, {\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 {1987}{10584 \, {\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 {2165}{222264 \, {\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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

9395/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2325/9604*x/sqrt(-10*x^2 - x + 3) + 53
95/57624/sqrt(-10*x^2 - x + 3) + 7625/12348*x/(-10*x^2 - x + 3)^(3/2) + 1/567/(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)) - 169/5292/(9*(-1
0*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)) + 1987/10584/(3*(-10*x^2
- x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 2165/222264/(-10*x^2 - x + 3)^(3/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\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)^(3/2)/((1 - 2*x)^(5/2)*(3*x + 2)^4),x)

[Out]

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

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sympy [F(-1)]  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)**(3/2)/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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