3.1960 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=108 \[ -\frac{125}{132} (1-2 x)^{11/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{5135}{756} (1-2 x)^{7/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{14}{729} (1-2 x)^{3/2}-\frac{98}{729} \sqrt{1-2 x}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-98*Sqrt[1 - 2*x])/729 - (14*(1 - 2*x)^(3/2))/729 - (2*(1 - 2*x)^(5/2))/405 - (5135*(1 - 2*x)^(7/2))/756 + (4
00*(1 - 2*x)^(9/2))/81 - (125*(1 - 2*x)^(11/2))/132 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Rubi [A]  time = 0.0387712, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ -\frac{125}{132} (1-2 x)^{11/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{5135}{756} (1-2 x)^{7/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{14}{729} (1-2 x)^{3/2}-\frac{98}{729} \sqrt{1-2 x}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-98*Sqrt[1 - 2*x])/729 - (14*(1 - 2*x)^(3/2))/729 - (2*(1 - 2*x)^(5/2))/405 - (5135*(1 - 2*x)^(7/2))/756 + (4
00*(1 - 2*x)^(9/2))/81 - (125*(1 - 2*x)^(11/2))/132 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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} (3+5 x)^3}{2+3 x} \, dx &=\int \left (\frac{5135}{108} (1-2 x)^{5/2}-\frac{400}{9} (1-2 x)^{7/2}+\frac{125}{12} (1-2 x)^{9/2}-\frac{(1-2 x)^{5/2}}{27 (2+3 x)}\right ) \, dx\\ &=-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}-\frac{1}{27} \int \frac{(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=-\frac{2}{405} (1-2 x)^{5/2}-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}-\frac{7}{81} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{14}{729} (1-2 x)^{3/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}-\frac{49}{243} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{98}{729} \sqrt{1-2 x}-\frac{14}{729} (1-2 x)^{3/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}-\frac{343}{729} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{98}{729} \sqrt{1-2 x}-\frac{14}{729} (1-2 x)^{3/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}+\frac{343}{729} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{98}{729} \sqrt{1-2 x}-\frac{14}{729} (1-2 x)^{3/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{5135}{756} (1-2 x)^{7/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{125}{132} (1-2 x)^{11/2}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0650082, size = 68, normalized size = 0.63 \[ \frac{\sqrt{1-2 x} \left (8505000 x^5+913500 x^4-7838550 x^3-249219 x^2+3024349 x-830656\right )}{280665}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-830656 + 3024349*x - 249219*x^2 - 7838550*x^3 + 913500*x^4 + 8505000*x^5))/280665 + (98*Sqrt[
7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Maple [A]  time = 0.005, size = 74, normalized size = 0.7 \begin{align*} -{\frac{14}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2}{405} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{5135}{756} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{400}{81} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{125}{132} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{98\,\sqrt{21}}{2187}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{98}{729}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-14/729*(1-2*x)^(3/2)-2/405*(1-2*x)^(5/2)-5135/756*(1-2*x)^(7/2)+400/81*(1-2*x)^(9/2)-125/132*(1-2*x)^(11/2)+9
8/2187*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-98/729*(1-2*x)^(1/2)

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Maxima [A]  time = 2.4977, size = 123, normalized size = 1.14 \begin{align*} -\frac{125}{132} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{400}{81} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{5135}{756} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2}{405} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{14}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{2187} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{98}{729} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/132*(-2*x + 1)^(11/2) + 400/81*(-2*x + 1)^(9/2) - 5135/756*(-2*x + 1)^(7/2) - 2/405*(-2*x + 1)^(5/2) - 14
/729*(-2*x + 1)^(3/2) - 49/2187*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 9
8/729*sqrt(-2*x + 1)

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Fricas [A]  time = 1.39618, size = 250, normalized size = 2.31 \begin{align*} \frac{49}{2187} \, \sqrt{7} \sqrt{3} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac{1}{280665} \,{\left (8505000 \, x^{5} + 913500 \, x^{4} - 7838550 \, x^{3} - 249219 \, x^{2} + 3024349 \, x - 830656\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/2187*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 1/280665*(8505000*x^5 + 9
13500*x^4 - 7838550*x^3 - 249219*x^2 + 3024349*x - 830656)*sqrt(-2*x + 1)

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Sympy [A]  time = 54.9393, size = 138, normalized size = 1.28 \begin{align*} - \frac{125 \left (1 - 2 x\right )^{\frac{11}{2}}}{132} + \frac{400 \left (1 - 2 x\right )^{\frac{9}{2}}}{81} - \frac{5135 \left (1 - 2 x\right )^{\frac{7}{2}}}{756} - \frac{2 \left (1 - 2 x\right )^{\frac{5}{2}}}{405} - \frac{14 \left (1 - 2 x\right )^{\frac{3}{2}}}{729} - \frac{98 \sqrt{1 - 2 x}}{729} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{729} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(11/2)/132 + 400*(1 - 2*x)**(9/2)/81 - 5135*(1 - 2*x)**(7/2)/756 - 2*(1 - 2*x)**(5/2)/405 - 14
*(1 - 2*x)**(3/2)/729 - 98*sqrt(1 - 2*x)/729 - 686*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))/729

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Giac [A]  time = 2.22051, size = 165, normalized size = 1.53 \begin{align*} \frac{125}{132} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{400}{81} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{5135}{756} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2}{405} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{14}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{2187} \, \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{98}{729} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/132*(2*x - 1)^5*sqrt(-2*x + 1) + 400/81*(2*x - 1)^4*sqrt(-2*x + 1) + 5135/756*(2*x - 1)^3*sqrt(-2*x + 1) -
2/405*(2*x - 1)^2*sqrt(-2*x + 1) - 14/729*(-2*x + 1)^(3/2) - 49/2187*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqr
t(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/729*sqrt(-2*x + 1)