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

Optimal. Leaf size=166 \[ -\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {5 x+3}}-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {6525}{392 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {37}{28 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}+\frac {2079585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]

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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 {36657025 \sqrt {1-2 x}}{332024 \sqrt {5 x+3}}-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {6525}{392 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {37}{28 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}+\frac {2079585 \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/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

-73435/(15092*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (36657025*Sqrt[1 - 2*x])/(332024*Sqrt[3 + 5*x]) + 1/(7*Sqrt[1 - 2
*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) + 37/(28*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 6525/(392*Sqrt[1 - 2*x]*(2
+ 3*x)*Sqrt[3 + 5*x]) + (2079585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*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 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}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{21} \int \frac {\frac {99}{2}-120 x}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{294} \int \frac {\frac {14595}{4}-11655 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {\int \frac {\frac {1198365}{8}-685125 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{2058}\\ &=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {-\frac {98442645}{16}+\frac {23132025 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{79233}\\ &=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {2 \int -\frac {5284225485}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{871563}\\ &=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {2079585 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {2079585 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {2079585 \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 = 0.09, size = 84, normalized size = 0.51 \begin {gather*} \frac {\frac {7 \left (1979479350 x^4+2925598635 x^3+622325745 x^2-723664682 x-283149136\right )}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}[phone] \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2324168} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(-283149136 - 723664682*x + 622325745*x^2 + 2925598635*x^3 + 1979479350*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^3*S
qrt[3 + 5*x]) + 251629785*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2324168

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IntegrateAlgebraic [A]  time = 0.28, size = 138, normalized size = 0.83 \begin {gather*} \frac {\sqrt {5 x+3} \left (-\frac {17150000 (1-2 x)^4}{(5 x+3)^4}-\frac {554420215 (1-2 x)^3}{(5 x+3)^3}-\frac {4697680680 (1-2 x)^2}{(5 x+3)^2}-\frac {12330147977 (1-2 x)}{5 x+3}+25088\right )}{332024 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )^3}+\frac {2079585 \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/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(Sqrt[3 + 5*x]*(25088 - (17150000*(1 - 2*x)^4)/(3 + 5*x)^4 - (554420215*(1 - 2*x)^3)/(3 + 5*x)^3 - (4697680680
*(1 - 2*x)^2)/(3 + 5*x)^2 - (12330147977*(1 - 2*x))/(3 + 5*x)))/(332024*Sqrt[1 - 2*x]*(7 + (1 - 2*x)/(3 + 5*x)
)^3) + (2079585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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fricas [A]  time = 1.28, size = 131, normalized size = 0.79 \begin {gather*} \frac {251629785 \, \sqrt {7} {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\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 (1979479350 \, x^{4} + 2925598635 \, x^{3} + 622325745 \, x^{2} - 723664682 \, x - 283149136\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4648336 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4648336*(251629785*sqrt(7)*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*arctan(1/14*sqrt(7)*(37*x + 2
0)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1979479350*x^4 + 2925598635*x^3 + 622325745*x^2 - 7236
64682*x - 283149136)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)

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giac [B]  time = 3.79, size = 403, normalized size = 2.43 \begin {gather*} -\frac {415917}{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 {625}{242} \, \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 )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1452605 \, {\left (2 \, x - 1\right )}} - \frac {297 \, {\left (37841 \, \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} + 16959040 \, \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} + 2009470400 \, \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 )}}{9604 \, {\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/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-415917/76832*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)))) - 625/242*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 64/1452605*sqrt(5)*sqrt(5*x +
3)*sqrt(-10*x + 5)/(2*x - 1) - 297/9604*(37841*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)))^5 + 16959040*sqrt(10)*((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 + 2009470400*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sq
rt(-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 = 1.84 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (67940041950 \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}-11574970110 \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}-25162978500 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-10131305548 \sqrt {-10 x^{2}-x +3}\, x -6039114840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3964087904 \sqrt {-10 x^{2}-x +3}\right )}{4648336 \left (3 x +2\right )^{3} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4648336*(-2*x+1)^(1/2)*(67940041950*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
8095*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^4-11574970110*7^(1/2)*x^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^3-25162978500*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^2-6039114840*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))-10131305548*(-10*x^2-x+3)^(1/2)*x-3964087904*(-10*x^2-x+3)^(1/2))/(3*x+2)^3/(2*x-1
)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.35, size = 211, normalized size = 1.27 \begin {gather*} -\frac {2079585}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {36657025 \, x}{166012 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {38272595}{332024 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{7 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {37}{28 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {6525}{392 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2079585/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 36657025/166012*x/sqrt(-10*x^2 - x
+ 3) - 38272595/332024/sqrt(-10*x^2 - x + 3) + 1/7/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^
2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 37/28/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x
^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 6525/392/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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