3.2115 \(\int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac{14517 \sqrt{1-2 x} (3 x+2)^3}{21175}+\frac{217152 \sqrt{1-2 x} (3 x+2)^2}{75625}+\frac{9 \sqrt{1-2 x} (1688625 x+5065808)}{378125}-\frac{402 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

[Out]

(217152*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 + (14517*Sqrt[1 - 2*x]*(2 + 3*x)^3)/21175 - (36*Sqrt[1 - 2*x]*(2 + 3*
x)^4)/(605*(3 + 5*x)) + (7*(2 + 3*x)^5)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(5065808 + 1688625*x))
/378125 - (402*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt[55])

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Rubi [A]  time = 0.0518739, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 153, 147, 63, 206} \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac{14517 \sqrt{1-2 x} (3 x+2)^3}{21175}+\frac{217152 \sqrt{1-2 x} (3 x+2)^2}{75625}+\frac{9 \sqrt{1-2 x} (1688625 x+5065808)}{378125}-\frac{402 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(217152*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 + (14517*Sqrt[1 - 2*x]*(2 + 3*x)^3)/21175 - (36*Sqrt[1 - 2*x]*(2 + 3*
x)^4)/(605*(3 + 5*x)) + (7*(2 + 3*x)^5)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(5065808 + 1688625*x))
/378125 - (402*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt[55])

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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}-\frac{1}{11} \int \frac{(2+3 x)^4 (243+417 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}-\frac{1}{605} \int \frac{(2+3 x)^3 (8670+14517 x)}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{14517 \sqrt{1-2 x} (2+3 x)^3}{21175}-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}+\frac{\int \frac{(-911757-1520064 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx}{21175}\\ &=\frac{217152 \sqrt{1-2 x} (2+3 x)^2}{75625}+\frac{14517 \sqrt{1-2 x} (2+3 x)^3}{21175}-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{(2+3 x) (63828618+106383375 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{529375}\\ &=\frac{217152 \sqrt{1-2 x} (2+3 x)^2}{75625}+\frac{14517 \sqrt{1-2 x} (2+3 x)^3}{21175}-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}+\frac{9 \sqrt{1-2 x} (5065808+1688625 x)}{378125}+\frac{201 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{378125}\\ &=\frac{217152 \sqrt{1-2 x} (2+3 x)^2}{75625}+\frac{14517 \sqrt{1-2 x} (2+3 x)^3}{21175}-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}+\frac{9 \sqrt{1-2 x} (5065808+1688625 x)}{378125}-\frac{201 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{378125}\\ &=\frac{217152 \sqrt{1-2 x} (2+3 x)^2}{75625}+\frac{14517 \sqrt{1-2 x} (2+3 x)^3}{21175}-\frac{36 \sqrt{1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)}+\frac{9 \sqrt{1-2 x} (5065808+1688625 x)}{378125}-\frac{402 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0937791, size = 101, normalized size = 0.72 \[ \frac{\frac{8820 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{\sqrt{1-2 x}}-\frac{55 \left (5011875 x^5+26663175 x^4+72309105 x^3+199582515 x^2-74439831 x-103960660\right )}{\sqrt{1-2 x} (5 x+3)}+546 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{13234375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(-103960660 - 74439831*x + 199582515*x^2 + 72309105*x^3 + 26663175*x^4 + 5011875*x^5))/(Sqrt[1 - 2*x]*(3
+ 5*x)) + 546*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] + (8820*Hypergeometric2F1[-1/2, 1, 1/2, (5*(1 - 2*x)
)/11])/Sqrt[1 - 2*x])/13234375

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Maple [A]  time = 0.01, size = 81, normalized size = 0.6 \begin{align*} -{\frac{729}{2800} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{2187}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{105057}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{315684}{3125}\sqrt{1-2\,x}}+{\frac{117649}{1936}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{1890625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{402\,\sqrt{55}}{20796875}{\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((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x)^2,x)

[Out]

-729/2800*(1-2*x)^(7/2)+2187/625*(1-2*x)^(5/2)-105057/5000*(1-2*x)^(3/2)+315684/3125*(1-2*x)^(1/2)+117649/1936
/(1-2*x)^(1/2)+2/1890625*(1-2*x)^(1/2)/(-2*x-6/5)-402/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.50078, size = 136, normalized size = 0.97 \begin{align*} -\frac{729}{2800} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2187}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{105057}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{201}{20796875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{315684}{3125} \, \sqrt{-2 \, x + 1} - \frac{1838265657 \, x + 1102959359}{3025000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/2800*(-2*x + 1)^(7/2) + 2187/625*(-2*x + 1)^(5/2) - 105057/5000*(-2*x + 1)^(3/2) + 201/20796875*sqrt(55)*
log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 315684/3125*sqrt(-2*x + 1) - 1/3025000*(18
38265657*x + 1102959359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 1.59664, size = 304, normalized size = 2.17 \begin{align*} \frac{1407 \, \sqrt{55}{\left (10 \, x^{2} + x - 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (55130625 \, x^{5} + 293294925 \, x^{4} + 795400155 \, x^{3} + 2195407665 \, x^{2} - 818846961 \, x - 1143572552\right )} \sqrt{-2 \, x + 1}}{145578125 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/145578125*(1407*sqrt(55)*(10*x^2 + x - 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(55130625*
x^5 + 293294925*x^4 + 795400155*x^3 + 2195407665*x^2 - 818846961*x - 1143572552)*sqrt(-2*x + 1))/(10*x^2 + x -
3)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.04965, size = 159, normalized size = 1.14 \begin{align*} \frac{729}{2800} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2187}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{105057}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{201}{20796875} \, \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{315684}{3125} \, \sqrt{-2 \, x + 1} - \frac{1838265657 \, x + 1102959359}{3025000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/2800*(2*x - 1)^3*sqrt(-2*x + 1) + 2187/625*(2*x - 1)^2*sqrt(-2*x + 1) - 105057/5000*(-2*x + 1)^(3/2) + 201
/20796875*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 315684/3125*s
qrt(-2*x + 1) - 1/3025000*(1838265657*x + 1102959359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))