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]

-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])

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Rubi [A]  time = 0.0588996, antiderivative size = 159, normalized size of antiderivative = 1., 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 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 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 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*}

Mathematica [A]  time = 0.0947818, 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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Maple [B]  time = 0.016, size = 305, normalized size = 1.9 \begin{align*}{\frac{1}{76697544\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 2184870773250\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-412697812725\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-757421868060\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+231950722640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-174789661860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -312356099510\,x\sqrt{-10\,{x}^{2}-x+3}-114725464168\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76697544*(1-2*x)^(1/2)*(2184870773250*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+4442570
572275*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2485897413120*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+891180391500*x^4*(-10*x^2-x+3)^(1/2)-412697812725*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1257704596050*x^3*(-10*x^2-x+3)^(1/2)-757421868060*7^(1/2)*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+231950722640*x^2*(-10*x^2-x+3)^(1/2)-174789661860*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-312356099510*x*(-10*x^2-x+3)^(1/2)-114725464168*(-10*x^2-x+3)^(1/2))/
(2+3*x)^2/(2*x-1)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.59709, size = 451, normalized size = 2.84 \begin{align*} -\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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 3.57594, size = 544, normalized size = 3.42 \begin{align*} -\frac{125}{63888} \, \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} + \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{11875}{2662} \, \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}}{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}} \end{align*}

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]

-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 + 243189/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 11875/2662*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))) - 64/228
2665*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) - 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)^2