3.1927 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=178 \[ \frac{117955 \sqrt{1-2 x}}{14 (5 x+3)}-\frac{176065 \sqrt{1-2 x}}{126 (5 x+3)^2}+\frac{1301 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{28 \sqrt{1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-176065*Sqrt[1 - 2*x])/(126*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^2) + (28*Sqrt[1 - 2*x])
/(3*(2 + 3*x)^2*(3 + 5*x)^2) + (1301*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + (117955*Sqrt[1 - 2*x])/(14*(3
+ 5*x)) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Rubi [A]  time = 0.0735551, antiderivative size = 178, 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 = {98, 151, 156, 63, 206} \[ \frac{117955 \sqrt{1-2 x}}{14 (5 x+3)}-\frac{176065 \sqrt{1-2 x}}{126 (5 x+3)^2}+\frac{1301 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{28 \sqrt{1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-176065*Sqrt[1 - 2*x])/(126*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^2) + (28*Sqrt[1 - 2*x])
/(3*(2 + 3*x)^2*(3 + 5*x)^2) + (1301*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + (117955*Sqrt[1 - 2*x])/(14*(3
+ 5*x)) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{1}{9} \int \frac{190-303 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1}{126} \int \frac{27202-41160 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac{1}{882} \int \frac{2963912-4098150 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{176065 \sqrt{1-2 x}}{126 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}-\frac{\int \frac{213252732-244026090 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{19404}\\ &=-\frac{176065 \sqrt{1-2 x}}{126 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac{117955 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{\int \frac{8809230276-5395025790 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{213444}\\ &=-\frac{176065 \sqrt{1-2 x}}{126 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac{117955 \sqrt{1-2 x}}{14 (3+5 x)}-\frac{1220574}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{564375}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{176065 \sqrt{1-2 x}}{126 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac{117955 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{1220574}{7} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{564375}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{176065 \sqrt{1-2 x}}{126 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{28 \sqrt{1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac{1301 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac{117955 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.12325, size = 104, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (15923925 x^4+40874010 x^3+39307638 x^2+16784696 x+2685098\right )}{14 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(2685098 + 16784696*x + 39307638*x^2 + 40874010*x^3 + 15923925*x^4))/(14*(2 + 3*x)^3*(3 + 5*x)^
2) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]]

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Maple [A]  time = 0.013, size = 103, normalized size = 0.6 \begin{align*} -324\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{3544\, \left ( 1-2\,x \right ) ^{5/2}}{21}}-{\frac{21418\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{25172\,\sqrt{1-2\,x}}{27}} \right ) }+{\frac{813716\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+2500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{269\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2937\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{112875\,\sqrt{55}}{11}{\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*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x)

[Out]

-324*(3544/21*(1-2*x)^(5/2)-21418/27*(1-2*x)^(3/2)+25172/27*(1-2*x)^(1/2))/(-6*x-4)^3+813716/49*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2500*(-269/20*(1-2*x)^(3/2)+2937/100*(1-2*x)^(1/2))/(-10*x-6)^2-112875/11*arcta
nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 4.1467, size = 221, normalized size = 1.24 \begin{align*} \frac{112875}{22} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{406858}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{15923925 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 145443720 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 498018162 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 757678432 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 432141633 \, \sqrt{-2 \, x + 1}}{7 \,{\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*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

112875/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 406858/49*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/7*(15923925*(-2*x + 1)^(9/2) - 145443720*(-2
*x + 1)^(7/2) + 498018162*(-2*x + 1)^(5/2) - 757678432*(-2*x + 1)^(3/2) + 432141633*sqrt(-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 = 1.66553, size = 587, normalized size = 3.3 \begin{align*} \frac{5530875 \, \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 ) + 8950876 \, \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 (15923925 \, x^{4} + 40874010 \, x^{3} + 39307638 \, x^{2} + 16784696 \, x + 2685098\right )} \sqrt{-2 \, x + 1}}{1078 \,{\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*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/1078*(5530875*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*sqrt(5)
*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 8950876*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 56
4*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(15923925*x^4 + 40874010*x^3 + 39307
638*x^2 + 16784696*x + 2685098)*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(-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((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.30867, size = 204, normalized size = 1.15 \begin{align*} \frac{112875}{22} \, \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{406858}{49} \, \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{25 \,{\left (1345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2937 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{3 \,{\left (15948 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 74963 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 88102 \, \sqrt{-2 \, x + 1}\right )}}{7 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

112875/22*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 406858/49*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 25/4*(1345*(-2*x + 1)^(3/2)
- 2937*sqrt(-2*x + 1))/(5*x + 3)^2 + 3/7*(15948*(2*x - 1)^2*sqrt(-2*x + 1) - 74963*(-2*x + 1)^(3/2) + 88102*s
qrt(-2*x + 1))/(3*x + 2)^3