3.1855 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=153 \[ \frac{34655 \sqrt{1-2 x}}{77 (5 x+3)}-\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)^2}+\frac{139 \sqrt{1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^2) + (139*Sqrt[1 - 2*x])/(14*(
2 + 3*x)*(3 + 5*x)^2) + (34655*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/7 - (66325*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.0595756, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {99, 151, 156, 63, 206} \[ \frac{34655 \sqrt{1-2 x}}{77 (5 x+3)}-\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)^2}+\frac{139 \sqrt{1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^2) + (139*Sqrt[1 - 2*x])/(14*(
2 + 3*x)*(3 + 5*x)^2) + (34655*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/7 - (66325*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}-\frac{1}{2} \int \frac{-23+35 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}-\frac{1}{14} \int \frac{-2513+3475 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac{1}{308} \int \frac{-180818+206910 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac{34655 \sqrt{1-2 x}}{77 (3+5 x)}-\frac{\int \frac{-7469374+4574460 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{3388}\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac{34655 \sqrt{1-2 x}}{77 (3+5 x)}-\frac{130401}{14} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{331625}{22} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac{34655 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{130401}{14} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{331625}{22} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac{139 \sqrt{1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac{34655 \sqrt{1-2 x}}{77 (3+5 x)}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0865959, size = 119, normalized size = 0.78 \[ \frac{77 \sqrt{1-2 x} \left (3118950 x^3+5926515 x^2+3748007 x+788875\right )[phone] \sqrt{21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-6499850 \sqrt{55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11858 (3 x+2)^2 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(77*Sqrt[1 - 2*x]*(788875 + 3748007*x + 5926515*x^2 + 3118950*x^3) + 10519014*Sqrt[21]*(6 + 19*x + 15*x^2)^2*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 6499850*Sqrt[55]*(6 + 19*x + 15*x^2)^2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1
1858*(2 + 3*x)^2*(3 + 5*x)^2)

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Maple [A]  time = 0.013, size = 94, normalized size = 0.6 \begin{align*} -972\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{209\, \left ( 1-2\,x \right ) ^{3/2}}{252}}-{\frac{211\,\sqrt{1-2\,x}}{108}} \right ) }+{\frac{43467\,\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{199\, \left ( 1-2\,x \right ) ^{3/2}}{220}}+{\frac{197\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{66325\,\sqrt{55}}{121}{\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)^(1/2)/(2+3*x)^3/(3+5*x)^3,x)

[Out]

-972*(209/252*(1-2*x)^(3/2)-211/108*(1-2*x)^(1/2))/(-6*x-4)^2+43467/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^
(1/2)+2500*(-199/220*(1-2*x)^(3/2)+197/100*(1-2*x)^(1/2))/(-10*x-6)^2-66325/121*arctanh(1/11*55^(1/2)*(1-2*x)^
(1/2))*55^(1/2)

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Maxima [A]  time = 1.67002, size = 197, normalized size = 1.29 \begin{align*} \frac{66325}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{43467}{98} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1559475 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 10604940 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 24027469 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 18137504 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66325/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 43467/98*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/77*(1559475*(-2*x + 1)^(7/2) - 10604940*(-2*x
+ 1)^(5/2) + 24027469*(-2*x + 1)^(3/2) - 18137504*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*
(2*x - 1)^2 + 20944*x - 4543)

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Fricas [A]  time = 1.60268, size = 510, normalized size = 3.33 \begin{align*} \frac{3249925 \, \sqrt{11} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5259507 \, \sqrt{7} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt{-2 \, x + 1}}{11858 \,{\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*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/11858*(3249925*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x +
1) + 5*x - 8)/(5*x + 3)) + 5259507*sqrt(7)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(-(sqrt(7)*s
qrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(3118950*x^3 + 5926515*x^2 + 3748007*x + 788875)*sqrt(-2*x +
1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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Sympy [A]  time = 121.286, size = 620, normalized size = 4.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3708*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(s
qrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 504*Pie
cewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqr
t(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1))
- 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) + 10100*Piecewise((sqrt(55)*(-log
(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1
)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (x <= 1/2) & (x > -3/5))) + 2200*Piecewise((sqrt(55)*(3*log(s
qrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11
+ 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*s
qrt(1 - 2*x)/11 - 1)**2))/6655, (x <= 1/2) & (x > -3/5))) - 18360*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 -
2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3)) + 30600*Piecewis
e((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11
)/55, 2*x - 1 > -11/5))

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Giac [A]  time = 2.03225, size = 200, normalized size = 1.31 \begin{align*} \frac{66325}{242} \, \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{43467}{98} \, \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{2 \,{\left (1559475 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 10604940 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 24027469 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 18137504 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66325/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 43467/98*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/77*(1559475*(2*x - 1)^3*sq
rt(-2*x + 1) + 10604940*(2*x - 1)^2*sqrt(-2*x + 1) - 24027469*(-2*x + 1)^(3/2) + 18137504*sqrt(-2*x + 1))/(15*
(2*x - 1)^2 + 136*x + 9)^2