3.22.15 \(\int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\) [2115]

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

[Out]

-402/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11*(2+3*x)^5/(3+5*x)/(1-2*x)^(1/2)+217152/75625*
(2+3*x)^2*(1-2*x)^(1/2)+14517/21175*(2+3*x)^3*(1-2*x)^(1/2)-36/605*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)+9/378125*(5
065808+1688625*x)*(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 158, 152, 65, 212} \begin {gather*} \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}} \end {gather*}

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 65

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 100

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 152

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 154

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] && ILtQ[m, -1] && GtQ[n, 0]

Rule 158

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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 \text {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*}

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Mathematica [A]
time = 0.13, size = 73, normalized size = 0.52 \begin {gather*} \frac {-\frac {55 \left (-1143572552-818846961 x+2195407665 x^2+795400155 x^3+293294925 x^4+55130625 x^5\right )}{\sqrt {1-2 x} (3+5 x)}-2814 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{145578125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(-1143572552 - 818846961*x + 2195407665*x^2 + 795400155*x^3 + 293294925*x^4 + 55130625*x^5))/(Sqrt[1 - 2
*x]*(3 + 5*x)) - 2814*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/145578125

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Maple [A]
time = 0.11, size = 81, normalized size = 0.58

method	result	size
risch	\(-\frac {55130625 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-818846961 x -1143572552}{2646875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {402 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}\)	\(61\)
derivativedivides	\(-\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{2800}+\frac {2187 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {105057 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {315684 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{1890625 \left (-\frac {6}{5}-2 x \right )}-\frac {402 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}+\frac {117649}{1936 \sqrt {1-2 x}}\)	\(81\)
default	\(-\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{2800}+\frac {2187 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {105057 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {315684 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{1890625 \left (-\frac {6}{5}-2 x \right )}-\frac {402 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}+\frac {117649}{1936 \sqrt {1-2 x}}\)	\(81\)
trager	\(\frac {\left (55130625 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-818846961 x -1143572552\right ) \sqrt {1-2 x}}{26468750 x^{2}+2646875 x -7940625}+\frac {201 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{20796875}\)	\(90\)

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,method=_RETURNVERBOSE)

[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)+2/1890625*(
1-2*x)^(1/2)/(-6/5-2*x)-402/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+117649/1936/(1-2*x)^(1/2)

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Maxima [A]
time = 0.48, size = 101, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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 = 1.06, size = 85, normalized size = 0.61 \begin {gather*} \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 {gather*}

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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Giac [A]
time = 1.49, size = 118, normalized size = 0.84 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.06, size = 84, normalized size = 0.60 \begin {gather*} \frac {315684\,\sqrt {1-2\,x}}{3125}-\frac {105057\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {2187\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {729\,{\left (1-2\,x\right )}^{7/2}}{2800}+\frac {\frac {1838265657\,x}{15125000}+\frac {1102959359}{15125000}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,402{}\mathrm {i}}{20796875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*402i)/20796875 + (315684*(1 - 2*x)^(1/2))/3125 - (105057*(1 -
2*x)^(3/2))/5000 + (2187*(1 - 2*x)^(5/2))/625 - (729*(1 - 2*x)^(7/2))/2800 + ((1838265657*x)/15125000 + 11029
59359/15125000)/((11*(1 - 2*x)^(1/2))/5 - (1 - 2*x)^(3/2))

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