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

Optimal. Leaf size=108 \[ -\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac {10}{63} (1-2 x)^{3/2} (27 x+22)+\frac {8}{9} \sqrt {1-2 x}-\frac {8}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 153, 147, 50, 63, 206} \begin {gather*} -\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac {10}{63} (1-2 x)^{3/2} (27 x+22)+\frac {8}{9} \sqrt {1-2 x}-\frac {8}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[1 - 2*x])/9 + (5*(1 - 2*x)^(3/2)*(3 + 5*x)^2)/7 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(3*(2 + 3*x)) - (10*(1
- 2*x)^(3/2)*(22 + 27*x))/63 - (8*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

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 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 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)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}+\frac {1}{3} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{2+3 x} \, dx\\ &=\frac {5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac {1}{63} \int \frac {(-288-810 x) \sqrt {1-2 x} (3+5 x)}{2+3 x} \, dx\\ &=\frac {5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac {10}{63} (1-2 x)^{3/2} (22+27 x)+\frac {4}{3} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {8}{9} \sqrt {1-2 x}+\frac {5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac {10}{63} (1-2 x)^{3/2} (22+27 x)+\frac {28}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {8}{9} \sqrt {1-2 x}+\frac {5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac {10}{63} (1-2 x)^{3/2} (22+27 x)-\frac {28}{9} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {8}{9} \sqrt {1-2 x}+\frac {5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac {10}{63} (1-2 x)^{3/2} (22+27 x)-\frac {8}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 70, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {1-2 x} \left (1500 x^4+780 x^3-1005 x^2-442 x+85\right )}{63 (3 x+2)}-\frac {8}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/63*(Sqrt[1 - 2*x]*(85 - 442*x - 1005*x^2 + 780*x^3 + 1500*x^4))/(2 + 3*x) - (8*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/9

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IntegrateAlgebraic [A]  time = 0.12, size = 90, normalized size = 0.83 \begin {gather*} \frac {\left (375 (1-2 x)^4-1890 (1-2 x)^3+2415 (1-2 x)^2+224 (1-2 x)-784\right ) \sqrt {1-2 x}}{126 (3 (1-2 x)-7)}-\frac {8}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-784 + 224*(1 - 2*x) + 2415*(1 - 2*x)^2 - 1890*(1 - 2*x)^3 + 375*(1 - 2*x)^4)*Sqrt[1 - 2*x])/(126*(-7 + 3*(1
- 2*x))) - (8*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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fricas [A]  time = 1.97, size = 80, normalized size = 0.74 \begin {gather*} \frac {28 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 3 \, {\left (1500 \, x^{4} + 780 \, x^{3} - 1005 \, x^{2} - 442 \, x + 85\right )} \sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/189*(28*sqrt(7)*sqrt(3)*(3*x + 2)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 3*(1500*x^4 +
780*x^3 - 1005*x^2 - 442*x + 85)*sqrt(-2*x + 1))/(3*x + 2)

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giac [A]  time = 1.02, size = 106, normalized size = 0.98 \begin {gather*} -\frac {125}{126} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {145}{54} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {10}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{27} \, \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 {214}{243} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/126*(2*x - 1)^3*sqrt(-2*x + 1) - 145/54*(2*x - 1)^2*sqrt(-2*x + 1) + 10/81*(-2*x + 1)^(3/2) + 4/27*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 214/243*sqrt(-2*x + 1) + 7/243
*sqrt(-2*x + 1)/(3*x + 2)

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maple [A]  time = 0.01, size = 72, normalized size = 0.67 \begin {gather*} -\frac {8 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{27}+\frac {125 \left (-2 x +1\right )^{\frac {7}{2}}}{126}-\frac {145 \left (-2 x +1\right )^{\frac {5}{2}}}{54}+\frac {10 \left (-2 x +1\right )^{\frac {3}{2}}}{81}+\frac {214 \sqrt {-2 x +1}}{243}-\frac {14 \sqrt {-2 x +1}}{729 \left (-2 x -\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/126*(-2*x+1)^(7/2)-145/54*(-2*x+1)^(5/2)+10/81*(-2*x+1)^(3/2)+214/243*(-2*x+1)^(1/2)-14/729*(-2*x+1)^(1/2)
/(-2*x-4/3)-8/27*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.32, size = 89, normalized size = 0.82 \begin {gather*} \frac {125}{126} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {145}{54} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {10}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{27} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {214}{243} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/126*(-2*x + 1)^(7/2) - 145/54*(-2*x + 1)^(5/2) + 10/81*(-2*x + 1)^(3/2) + 4/27*sqrt(21)*log(-(sqrt(21) - 3
*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 214/243*sqrt(-2*x + 1) + 7/243*sqrt(-2*x + 1)/(3*x + 2)

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mupad [B]  time = 1.19, size = 73, normalized size = 0.68 \begin {gather*} \frac {14\,\sqrt {1-2\,x}}{729\,\left (2\,x+\frac {4}{3}\right )}+\frac {214\,\sqrt {1-2\,x}}{243}+\frac {10\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {145\,{\left (1-2\,x\right )}^{5/2}}{54}+\frac {125\,{\left (1-2\,x\right )}^{7/2}}{126}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,8{}\mathrm {i}}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*8i)/27 + (14*(1 - 2*x)^(1/2))/(729*(2*x + 4/3)) + (214*(1 - 2*
x)^(1/2))/243 + (10*(1 - 2*x)^(3/2))/81 - (145*(1 - 2*x)^(5/2))/54 + (125*(1 - 2*x)^(7/2))/126

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sympy [F(-1)]  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((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**2,x)

[Out]

Timed out

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