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

Optimal. Leaf size=100 \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac {5 \sqrt {1-2 x} (2815 x+323)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {97, 149, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac {5 \sqrt {1-2 x} (2815 x+323)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x
]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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 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 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} (3+5 x)^3}{(2+3 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{126} \int \frac {(643-2815 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}-\frac {7559 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1134}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 63, normalized size = 0.63 \begin {gather*} \frac {\sqrt {1-2 x} \left (31500 x^3+7350 x^2-32833 x-15815\right )}{1134 (3 x+2)^2}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-15815 - 32833*x + 7350*x^2 + 31500*x^3))/(1134*(2 + 3*x)^2) + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(567*Sqrt[21])

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IntegrateAlgebraic [A]  time = 0.20, size = 79, normalized size = 0.79 \begin {gather*} \frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}-\frac {\left (7875 (1-2 x)^3-27300 (1-2 x)^2-1858 (1-2 x)+52913\right ) \sqrt {1-2 x}}{567 (3 (1-2 x)-7)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/567*((52913 - 1858*(1 - 2*x) - 27300*(1 - 2*x)^2 + 7875*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(-7 + 3*(1 - 2*x))^2 +
(7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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fricas [A]  time = 1.42, size = 80, normalized size = 0.80 \begin {gather*} \frac {7559 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (31500 \, x^{3} + 7350 \, x^{2} - 32833 \, x - 15815\right )} \sqrt {-2 \, x + 1}}{23814 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23814*(7559*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31500*x^3 +
7350*x^2 - 32833*x - 15815)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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giac [A]  time = 1.36, size = 86, normalized size = 0.86 \begin {gather*} -\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \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 {50}{27} \, \sqrt {-2 \, x + 1} + \frac {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{2268 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1/2268*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(3*x + 2)^2

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maple [A]  time = 0.01, size = 66, normalized size = 0.66 \begin {gather*} \frac {7559 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{11907}-\frac {125 \left (-2 x +1\right )^{\frac {3}{2}}}{81}-\frac {50 \sqrt {-2 x +1}}{27}-\frac {2 \left (-\frac {211 \left (-2 x +1\right )^{\frac {3}{2}}}{126}+\frac {209 \sqrt {-2 x +1}}{54}\right )}{3 \left (-6 x -4\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/81*(-2*x+1)^(3/2)-50/27*(-2*x+1)^(1/2)-2/3*(-211/126*(-2*x+1)^(3/2)+209/54*(-2*x+1)^(1/2))/(-6*x-4)^2+755
9/11907*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.22, size = 92, normalized size = 0.92 \begin {gather*} -\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {50}{27} \, \sqrt {-2 \, x + 1} + \frac {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{567 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 50/27*sqrt(-2*x + 1) + 1/567*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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mupad [B]  time = 0.07, size = 74, normalized size = 0.74 \begin {gather*} -\frac {50\,\sqrt {1-2\,x}}{27}-\frac {125\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {209\,\sqrt {1-2\,x}}{729}-\frac {211\,{\left (1-2\,x\right )}^{3/2}}{1701}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,7559{}\mathrm {i}}{11907} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*7559i)/11907 - (50*(1 - 2*x)^(1/2))/27 - (125*(1 - 2*x)^(3/2
))/81 - ((209*(1 - 2*x)^(1/2))/729 - (211*(1 - 2*x)^(3/2))/1701)/((28*x)/3 + (2*x - 1)^2 + 7/9)

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

[Out]

Timed out

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