3.1978 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=153 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5) + (63*Sqrt[1 - 2*x])/(10*(2 + 3*x)^4) + (1201*Sqrt[1 - 2*x])/(30*(2 + 3*x
)^3) + (25159*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2) + (584179*Sqrt[1 - 2*x])/(196*(2 + 3*x)) + (20149879*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(98*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0687126, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5) + (63*Sqrt[1 - 2*x])/(10*(2 + 3*x)^4) + (1201*Sqrt[1 - 2*x])/(30*(2 + 3*x
)^3) + (25159*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2) + (584179*Sqrt[1 - 2*x])/(196*(2 + 3*x)) + (20149879*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(98*Sqrt[21]) - 6050*Sqrt[55]*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{(228-225 x) \sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}-\frac{1}{180} \int \frac{-25182+37890 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}-\frac{\int \frac{-2761290+3783150 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}-\frac{\int \frac{-209531070+237752550 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}-\frac{\int \frac{-9014096070+5520491550 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{370440}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}-\frac{20149879}{196} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+166375 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}+\frac{20149879}{196} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-166375 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac{63 \sqrt{1-2 x}}{10 (2+3 x)^4}+\frac{1201 \sqrt{1-2 x}}{30 (2+3 x)^3}+\frac{25159 \sqrt{1-2 x}}{84 (2+3 x)^2}+\frac{584179 \sqrt{1-2 x}}{196 (2+3 x)}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.125893, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \left (709777485 x^4+1916515215 x^3+1941349752 x^2+874383298 x+147756688\right )}{2940 (3 x+2)^5}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(147756688 + 874383298*x + 1941349752*x^2 + 1916515215*x^3 + 709777485*x^4))/(2940*(2 + 3*x)^5)
+ (20149879*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(98*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.011, size = 93, normalized size = 0.6 \begin{align*} -486\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{584179\, \left ( 1-2\,x \right ) ^{9/2}}{588}}-{\frac{504319\, \left ( 1-2\,x \right ) ^{7/2}}{54}}+{\frac{13335122\, \left ( 1-2\,x \right ) ^{5/2}}{405}}-{\frac{75232787\, \left ( 1-2\,x \right ) ^{3/2}}{1458}}+{\frac{29479429\,\sqrt{1-2\,x}}{972}} \right ) }+{\frac{20149879\,\sqrt{21}}{2058}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-6050\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x),x)

[Out]

-486*(584179/588*(1-2*x)^(9/2)-504319/54*(1-2*x)^(7/2)[phone]/405*(1-2*x)^(5/2)-75232787/1458*(1-2*x)^(3/2)+
29479429/972*(1-2*x)^(1/2))/(-6*x-4)^5+20149879/2058*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6050*arctanh
(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 3.82177, size = 221, normalized size = 1.44 \begin{align*} 3025 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20149879}{4116} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{709777485 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 6672140370 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 23523155208 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{1470 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x),x, algorithm="maxima")

[Out]

3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20149879/4116*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1470*(709777485*(-2*x + 1)^(9/2) - 6672140370
*(-2*x + 1)^(7/2) + 23523155208*(-2*x + 1)^(5/2) - 36864065630*(-2*x + 1)^(3/2) + 21667380315*sqrt(-2*x + 1))/
(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 1.3695, size = 552, normalized size = 3.61 \begin{align*} \frac{62254500 \, \sqrt{55}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 100749395 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \,{\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt{-2 \, x + 1}}{20580 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/20580*(62254500*sqrt(55)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((5*x + sqrt(55)*sqrt(-2*x
+ 1) - 8)/(5*x + 3)) + 100749395*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sq
rt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 7*(709777485*x^4 + 1916515215*x^3 + 1941349752*x^2 + 874383298*x + 147
756688)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**(5/2)/(2+3*x)**6/(3+5*x),x)

[Out]

Timed out

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Giac [A]  time = 1.70704, size = 209, normalized size = 1.37 \begin{align*} 3025 \, \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{20149879}{4116} \, \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{709777485 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 6672140370 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 23523155208 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{47040 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x),x, algorithm="giac")

[Out]

3025*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20149879/4116*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/47040*(709777485*(2*x - 1)
^4*sqrt(-2*x + 1) + 6672140370*(2*x - 1)^3*sqrt(-2*x + 1) + 23523155208*(2*x - 1)^2*sqrt(-2*x + 1) - 368640656
30*(-2*x + 1)^(3/2) + 21667380315*sqrt(-2*x + 1))/(3*x + 2)^5