3.2068 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-2076675*Sqrt[1 - 2*x])/(7546*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)^2) + (90*Sqrt[1 - 2*x])/(
49*(2 + 3*x)^2*(3 + 5*x)^2) + (12555*Sqrt[1 - 2*x])/(343*(2 + 3*x)*(3 + 5*x)^2) + (137735775*Sqrt[1 - 2*x])/(8
3006*(3 + 5*x)) + (7852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0713214, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {103, 151, 156, 63, 206} \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2076675*Sqrt[1 - 2*x])/(7546*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)^2) + (90*Sqrt[1 - 2*x])/(
49*(2 + 3*x)^2*(3 + 5*x)^2) + (12555*Sqrt[1 - 2*x])/(343*(2 + 3*x)*(3 + 5*x)^2) + (137735775*Sqrt[1 - 2*x])/(8
3006*(3 + 5*x)) + (7852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/121

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 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{1}{21} \int \frac{90-135 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{1}{294} \int \frac{12510-18900 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{\int \frac{1362060-1883250 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx}{2058}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}-\frac{\int \frac{97998660-112140450 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{45276}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{\int \frac{4048216380-2479243950 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{498036}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}-\frac{11779020}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{13449375}{242} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{11779020}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{13449375}{242} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0988242, size = 106, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \left (18594329625 x^4+47728484550 x^3+45899434890 x^2+19599448500 x+3135381218\right )}{83006 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(3135381218 + 19599448500*x + 45899434890*x^2 + 47728484550*x^3 + 18594329625*x^4))/(83006*(2 +
3*x)^3*(3 + 5*x)^2) + (7852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.012, size = 103, normalized size = 0.6 \begin{align*} -2916\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{3755\, \left ( 1-2\,x \right ) ^{5/2}}{1029}}-{\frac{22690\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{3809\,\sqrt{1-2\,x}}{189}} \right ) }+{\frac{7852680\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+312500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{261\, \left ( 1-2\,x \right ) ^{3/2}}{12100}}+{\frac{259\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{2689875\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2916*(3755/1029*(1-2*x)^(5/2)-22690/1323*(1-2*x)^(3/2)+3809/189*(1-2*x)^(1/2))/(-6*x-4)^3+7852680/2401*arctan
h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+312500*(-261/12100*(1-2*x)^(3/2)+259/5500*(1-2*x)^(1/2))/(-10*x-6)^2-26
89875/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.8111, size = 221, normalized size = 1.23 \begin{align*} \frac{2689875}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3926340}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18594329625 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 169834287600 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 581534624610 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 884739292920 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 504610725773 \, \sqrt{-2 \, x + 1}}{41503 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2689875/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3926340/2401*sqrt(21
)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/41503*(18594329625*(-2*x + 1)^(9/2) -
169834287600*(-2*x + 1)^(7/2) + 581534624610*(-2*x + 1)^(5/2) - 884739292920*(-2*x + 1)^(3/2) + 504610725773*s
qrt(-2*x + 1))/(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)

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Fricas [A]  time = 1.86304, size = 621, normalized size = 3.45 \begin{align*} \frac{6458389875 \, \sqrt{11} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 10451917080 \, \sqrt{7} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (18594329625 \, x^{4} + 47728484550 \, x^{3} + 45899434890 \, x^{2} + 19599448500 \, x + 3135381218\right )} \sqrt{-2 \, x + 1}}{6391462 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6391462*(6458389875*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*s
qrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 10451917080*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 176
6*x^2 + 564*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(18594329625*x^4 + 4772848
4550*x^3 + 45899434890*x^2 + 19599448500*x + 3135381218)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766
*x^2 + 564*x + 72)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.42145, size = 204, normalized size = 1.13 \begin{align*} \frac{2689875}{2662} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3926340}{2401} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{625 \,{\left (1305 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2849 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} + \frac{27 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 158830 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 186641 \, \sqrt{-2 \, x + 1}\right )}}{686 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2689875/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3926340/24
01*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/484*(1305*(-2*x +
1)^(3/2) - 2849*sqrt(-2*x + 1))/(5*x + 3)^2 + 27/686*(33795*(2*x - 1)^2*sqrt(-2*x + 1) - 158830*(-2*x + 1)^(3
/2) + 186641*sqrt(-2*x + 1))/(3*x + 2)^3