3.2515 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{5 x+3}}-\frac{207895 \sqrt{1-2 x}}{6468 (5 x+3)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

[Out]

(-207895*Sqrt[1 - 2*x])/(6468*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (753*Sqr
t[1 - 2*x])/(196*(2 + 3*x)*(3 + 5*x)^(3/2)) + (20743985*Sqrt[1 - 2*x])/(71148*Sqrt[3 + 5*x]) - (392283*ArcTan[
Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Rubi [A]  time = 0.044746, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{5 x+3}}-\frac{207895 \sqrt{1-2 x}}{6468 (5 x+3)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-207895*Sqrt[1 - 2*x])/(6468*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (753*Sqr
t[1 - 2*x])/(196*(2 + 3*x)*(3 + 5*x)^(3/2)) + (20743985*Sqrt[1 - 2*x])/(71148*Sqrt[3 + 5*x]) - (392283*ArcTan[
Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*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}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{1}{14} \int \frac{\frac{131}{2}-90 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac{1}{98} \int \frac{\frac{23507}{4}-7530 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{207895 \sqrt{1-2 x}}{6468 (3+5 x)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{\frac{2651953}{8}-\frac{623685 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{1617}\\ &=-\frac{207895 \sqrt{1-2 x}}{6468 (3+5 x)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{3+5 x}}+\frac{2 \int \frac{142398729}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{17787}\\ &=-\frac{207895 \sqrt{1-2 x}}{6468 (3+5 x)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{3+5 x}}+\frac{392283}{392} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{207895 \sqrt{1-2 x}}{6468 (3+5 x)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{3+5 x}}+\frac{392283}{196} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{207895 \sqrt{1-2 x}}{6468 (3+5 x)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{3+5 x}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{196 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0629895, size = 79, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (933479325 x^3+1784145090 x^2+1135041037 x+240342364\right )}{71148 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(240342364 + 1135041037*x + 1784145090*x^2 + 933479325*x^3))/(71148*(2 + 3*x)^2*(3 + 5*x)^(3/2)
) - (392283*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Maple [B]  time = 0.016, size = 250, normalized size = 1.8 \begin{align*}{\frac{1}{996072\, \left ( 2+3\,x \right ) ^{2}} \left ( 32039714025\,\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}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+24978031260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\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/(2+3*x)^3/(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

1/996072*(32039714025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+81167275530*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+77037712389*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))*x^2+13068710550*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x+24978031260*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-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} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54993, size = 392, normalized size = 2.86 \begin{align*} -\frac{142398729 \, \sqrt{7}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, 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 (933479325 \, x^{3} + 1784145090 \, x^{2} + 1135041037 \, x + 240342364\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{996072 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/996072*(142398729*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5
*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(933479325*x^3 + 1784145090*x^2 + 1135041037*x + 240342364)*sqrt
(5*x + 3)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.48989, size = 509, normalized size = 3.72 \begin{align*} -\frac{25}{5808} \, \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{392283}{27440} \, \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{2425}{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{297 \,{\left (461 \, \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} + 110600 \, \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/(2+3*x)^3/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-25/5808*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 + 392283/27440*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)))) + 2425/242*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))) + 297/98*(46
1*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) - sq
rt(22)))^3 + 110600*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^2