3.2032 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=80 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac{5 \sqrt{1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac{78710 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x]*(1205 + 1867*x))/(9261*(2 + 3*x)^2) - (78710*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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Rubi [A]  time = 0.0183083, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {98, 145, 63, 206} \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac{5 \sqrt{1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac{78710 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x]*(1205 + 1867*x))/(9261*(2 + 3*x)^2) - (78710*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^4} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)^3}-\frac{1}{63} \int \frac{(-290-520 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac{5 \sqrt{1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}+\frac{39355 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{9261}\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac{5 \sqrt{1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}-\frac{39355 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{9261}\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac{5 \sqrt{1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}-\frac{78710 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.03865, size = 58, normalized size = 0.72 \[ \frac{\frac{21 \sqrt{1-2 x} \left (31680 x^2+41155 x+13373\right )}{(3 x+2)^3}-78710 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{194481} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]

[Out]

((21*Sqrt[1 - 2*x]*(13373 + 41155*x + 31680*x^2))/(2 + 3*x)^3 - 78710*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/194481

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Maple [A]  time = 0.008, size = 57, normalized size = 0.7 \begin{align*} 54\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3520\, \left ( 1-2\,x \right ) ^{5/2}}{27783}}+{\frac{20810\, \left ( 1-2\,x \right ) ^{3/2}}{35721}}-{\frac{3418\,\sqrt{1-2\,x}}{5103}} \right ) }-{\frac{78710\,\sqrt{21}}{194481}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x)

[Out]

54*(-3520/27783*(1-2*x)^(5/2)+20810/35721*(1-2*x)^(3/2)-3418/5103*(1-2*x)^(1/2))/(-6*x-4)^3-78710/194481*arcta
nh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.58139, size = 124, normalized size = 1.55 \begin{align*} \frac{39355}{194481} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (15840 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 72835 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83741 \, \sqrt{-2 \, x + 1}\right )}}{9261 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

39355/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/9261*(15840*(-2*x
+ 1)^(5/2) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 1.6365, size = 251, normalized size = 3.14 \begin{align*} \frac{39355 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (31680 \, x^{2} + 41155 \, x + 13373\right )} \sqrt{-2 \, x + 1}}{194481 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/194481*(39355*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*
(31680*x^2 + 41155*x + 13373)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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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((3+5*x)**3/(2+3*x)**4/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.48962, size = 113, normalized size = 1.41 \begin{align*} \frac{39355}{194481} \, \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{15840 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 72835 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83741 \, \sqrt{-2 \, x + 1}}{18522 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

39355/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/18522*(15
840*(2*x - 1)^2*sqrt(-2*x + 1) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(3*x + 2)^3