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

Optimal. Leaf size=166 \[ \frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {5 x+3}}-\frac {21891025 \sqrt {1-2 x}}{90552 (5 x+3)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]

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Rubi [A]  time = 0.06, antiderivative size = 166, 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 = {103, 151, 152, 12, 93, 204} \begin {gather*} \frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {5 x+3}}-\frac {21891025 \sqrt {1-2 x}}{90552 (5 x+3)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-21891025*Sqrt[1 - 2*x])/(90552*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (325*Sqrt[
1 - 2*x])/(196*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (79335*Sqrt[1 - 2*x])/(2744*(2 + 3*x)*(3 + 5*x)^(3/2)) + (218436
9575*Sqrt[1 - 2*x])/(996072*Sqrt[3 + 5*x]) - (41307885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sq
rt[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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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 {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {1}{21} \int \frac {\frac {165}{2}-120 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{294} \int \frac {\frac {40335}{4}-14625 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {\frac {7422495}{8}-1190025 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{2058}\\ &=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {\frac {837775605}{16}-\frac {197019225 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{33957}\\ &=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {2 \int \frac {44984286765}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{373527}\\ &=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {41307885 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {41307885 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 84, normalized size = 0.51 \begin {gather*} \frac {\sqrt {1-2 x} \left (294889892625 x^4+760212086400 x^3+734310313245 x^2+314968389410 x+50617099616\right )}{996072 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(50617099616 + 314968389410*x + 734310313245*x^2 + 760212086400*x^3 + 294889892625*x^4))/(99607
2*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (41307885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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IntegrateAlgebraic [A]  time = 0.27, size = 157, normalized size = 0.95 \begin {gather*} \frac {-\frac {17150000 (1-2 x)^{9/2}}{(5 x+3)^{9/2}}+\frac {977550000 (1-2 x)^{7/2}}{(5 x+3)^{7/2}}+\frac {32987507745 (1-2 x)^{5/2}}{(5 x+3)^{5/2}}+\frac {279901897240 (1-2 x)^{3/2}}{(5 x+3)^{3/2}}+\frac {734743275231 \sqrt {1-2 x}}{\sqrt {5 x+3}}}{996072 \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-17150000*(1 - 2*x)^(9/2))/(3 + 5*x)^(9/2) + (977550000*(1 - 2*x)^(7/2))/(3 + 5*x)^(7/2) + (32987507745*(1 -
2*x)^(5/2))/(3 + 5*x)^(5/2) + (279901897240*(1 - 2*x)^(3/2))/(3 + 5*x)^(3/2) + (734743275231*Sqrt[1 - 2*x])/S
qrt[3 + 5*x])/(996072*(7 + (1 - 2*x)/(3 + 5*x))^3) - (41307885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(2744*Sqrt[7])

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fricas [A]  time = 1.35, size = 131, normalized size = 0.79 \begin {gather*} -\frac {14994762255 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 (294889892625 \, x^{4} + 760212086400 \, x^{3} + 734310313245 \, x^{2} + 314968389410 \, x + 50617099616\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{13945008 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13945008*(14994762255*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(
37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(294889892625*x^4 + 760212086400*x^3 + 73431031
3245*x^2 + 314968389410*x + 50617099616)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x
^2 + 564*x + 72)

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giac [B]  time = 3.21, size = 434, normalized size = 2.61 \begin {gather*} \frac {8261577}{76832} \, \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 {125}{5808} \, \sqrt {10} {\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 )}^{3} - \frac {3120 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {12480 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {1485 \, {\left (13759 \, \sqrt {10} {\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} + 6614720 \, \sqrt {10} {\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} + 818950720 \, \sqrt {10} {\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 )}\right )}}{1372 \, {\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+3*x)^4/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

8261577/76832*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)))) - 125/5808*sqrt(10)*(((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 - 3120*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) + 12480*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 1485/1372*(13759*s
qrt(10)*((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 + 6614720*sqrt(10)*((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 + 818950720*sqrt(10)*((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))))/(((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 = 298, normalized size = 1.80 \begin {gather*} \frac {\left (10121464522125 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{13945008 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13945008*(10121464522125*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/
2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+26480750142330*7^(1/2)*x^2*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+8457045911820*7^(1/2)*x*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+1079622882360*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+708639394624*(-10*x^2-x+3)^(1/2))*(-
2*x+1)^(1/2)/(3*x+2)^3/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{4} \sqrt {-2 \, x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(1 - 2*x)*(3*x + 2)**4*(5*x + 3)**(5/2)), x)

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