3.2578 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

-58/(539*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (28705*Sqrt[1 - 2*x])/(17787*(3 + 5*x)^(3/2)) + 3/(7*Sqrt[1 - 2*x]*(
2 + 3*x)*(3 + 5*x)^(3/2)) + (2841815*Sqrt[1 - 2*x])/(195657*Sqrt[3 + 5*x]) - (4887*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.0470482, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-58/(539*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (28705*Sqrt[1 - 2*x])/(17787*(3 + 5*x)^(3/2)) + 3/(7*Sqrt[1 - 2*x]*(
2 + 3*x)*(3 + 5*x)^(3/2)) + (2841815*Sqrt[1 - 2*x])/(195657*Sqrt[3 + 5*x]) - (4887*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(49*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 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)^2 (3+5 x)^{5/2}} \, dx &=\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{1}{7} \int \frac{\frac{61}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac{2}{539} \int \frac{-\frac{3653}{4}+870 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{4 \int \frac{-\frac{361687}{8}+\frac{86115 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{17787}\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}-\frac{8 \int -\frac{19513791}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{195657}\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}+\frac{4887}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}+\frac{4887}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0607669, size = 102, normalized size = 0.78 \[ \frac{-7 \left (85254450 x^3+63467215 x^2-20145298 x-16461125\right )-19513791 \sqrt{7-14 x} \sqrt{5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1369599 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(-16461125 - 20145298*x + 63467215*x^2 + 85254450*x^3) - 19513791*Sqrt[7 - 14*x]*Sqrt[3 + 5*x]*(6 + 19*x +
15*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1369599*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2))

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Maple [B]  time = 0.016, size = 257, normalized size = 2. \begin{align*}{\frac{1}{ \left ( 5478396+8217594\,x \right ) \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 2927068650\,\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]\,\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}-995203341\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+888541010\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-351248238\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -282034172\,x\sqrt{-10\,{x}^{2}-x+3}-230455750\,\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)^2/(3+5*x)^(5/2),x)

[Out]

1/2739198*(1-2*x)^(1/2)*(2927068650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+4000327155*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+663468894*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^2+1193562300*x^3*(-10*x^2-x+3)^(1/2)-995203341*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x+888541010*x^2*(-10*x^2-x+3)^(1/2)-351248238*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))-282034172*x*(-10*x^2-x+3)^(1/2)-230455750*(-10*x^2-x+3)^(1/2))/(2+3*x)/(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 )}^{2}{\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)^2/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57356, size = 378, normalized size = 2.91 \begin{align*} -\frac{19513791 \, \sqrt{7}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\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 (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2739198 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2739198*(19513791*sqrt(7)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(85254450*x^3 + 63467215*x^2 - 20145298*x - 16461125)*sqrt(5*x + 3
)*sqrt(-2*x + 1))/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.08536, size = 458, normalized size = 3.52 \begin{align*} -\frac{25}{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{4887}{6860} \, \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{775}{1331} \, \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{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{326095 \,{\left (2 \, x - 1\right )}} + \frac{1782 \, \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 )}}{49 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/63888*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 + 4887/6860*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)))) + 775/1331*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))) - 32/326095*sq
rt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1782/49*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)