3.2579 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=159 \[ \frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {5 x+3}}-\frac {7090175 \sqrt {1-2 x}}{498036 (5 x+3)^{3/2}}-\frac {8515}{7546 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-\frac {1215945 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]

[Out]

-1215945/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-8515/7546/(3+5*x)^(3/2)/(1-2*x)^(1/2)+3/
14/(2+3*x)^2/(3+5*x)^(3/2)/(1-2*x)^(1/2)+765/196/(2+3*x)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-7090175/498036*(1-2*x)^(1
/2)/(3+5*x)^(3/2)[phone]/5478396*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 159, 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} \[ \frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {5 x+3}}-\frac {7090175 \sqrt {1-2 x}}{498036 (5 x+3)^{3/2}}-\frac {8515}{7546 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-\frac {1215945 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-8515/(7546*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (7090175*Sqrt[1 - 2*x])/(498036*(3 + 5*x)^(3/2)) + 3/(14*Sqrt[1 -
2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 765/(196*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (707286025*Sqrt[1 - 2
*x])/(5478396*Sqrt[3 + 5*x]) - (1215945*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*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)^3 (3+5 x)^{5/2}} \, dx &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{14} \int \frac {\frac {95}{2}-120 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{98} \int \frac {\frac {14435}{4}-11475 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {804955}{8}+127725 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{3773}\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2 \int \frac {-\frac {90407945}{16}+\frac {21270525 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{124509}\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {3+5 x}}-\frac {4 \int -\frac {4855268385}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1369599}\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {3+5 x}}+\frac {1215945 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {3+5 x}}+\frac {1215945 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {3+5 x}}-\frac {1215945 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 84, normalized size = 0.53 \[ \frac {-\frac {7 \left (63655742250 x^4+89836042575 x^3+16567908760 x^2-22311149965 x-8194676012\right )}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-4855268385 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{38348772} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-8194676012 - 22311149965*x + 16567908760*x^2 + 89836042575*x^3 + 63655742250*x^4))/(Sqrt[1 - 2*x]*(2 +
3*x)^2*(3 + 5*x)^(3/2)) - 4855268385*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/38348772

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fricas [A]  time = 0.75, size = 131, normalized size = 0.82 \[ -\frac {4855268385 \, \sqrt {7} {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (63655742250 \, x^{4} + 89836042575 \, x^{3} + 16567908760 \, x^{2} - 22311149965 \, x - 8194676012\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{76697544 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76697544*(4855268385*sqrt(7)*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(63655742250*x^4 + 89836042575*x^3 + 16567908760*x^2
- 22311149965*x - 8194676012)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 3
6)

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giac [B]  time = 3.19, size = 399, normalized size = 2.51 \[ \frac {243189}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{63888} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {9120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2282665 \, {\left (2 \, x - 1\right )}} + \frac {891 \, {\left (67 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 16120 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243189/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 125/63888*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 - 2280*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) + 9120*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 64/2282665*sqrt(5)*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 891/98*(67*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 + 16120*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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maple [B]  time = 0.02, size = 305, normalized size = 1.92 \[ \frac {\sqrt {-2 x +1}\, \left (2184870773250 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}-412697812725 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-757421868060 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-312356099510 \sqrt {-10 x^{2}-x +3}\, x -174789661860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-114725464168 \sqrt {-10 x^{2}-x +3}\right )}{76697544 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76697544*(-2*x+1)^(1/2)*(2184870773250*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+444257
0572275*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^4-412697812725*7^(1/2)*x^2*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3-757421868060*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^2-174789661860*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-312356099510*(-10*x^2-x+3)^(1/2)*x-114725464168*(-10*x^2-x+3)^(1/2))
/(3*x+2)^2/(2*x-1)/(-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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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