3.1951 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=121 \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

[Out]

(24965*Sqrt[1 - 2*x])/15876 - (1 - 2*x)^(7/2)/(252*(2 + 3*x)^4) + (31*(1 - 2*x)^(7/2))/(588*(2 + 3*x)^3) - (49
93*(1 - 2*x)^(5/2))/(10584*(2 + 3*x)^2) + (24965*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)) - (24965*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

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Rubi [A]  time = 0.0330275, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 50, 63, 206} \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24965*Sqrt[1 - 2*x])/15876 - (1 - 2*x)^(7/2)/(252*(2 + 3*x)^4) + (31*(1 - 2*x)^(7/2))/(588*(2 + 3*x)^3) - (49
93*(1 - 2*x)^(5/2))/(10584*(2 + 3*x)^2) + (24965*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)) - (24965*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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)^2}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{1}{252} \int \frac{(1-2 x)^{5/2} (1121+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}+\frac{4993 \int \frac{(1-2 x)^{5/2}}{(2+3 x)^3} \, dx}{1764}\\ &=-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac{4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}-\frac{24965 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac{4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac{24965 \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx}{10584}\\ &=\frac{24965 \sqrt{1-2 x}}{15876}-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac{4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac{24965 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{4536}\\ &=\frac{24965 \sqrt{1-2 x}}{15876}-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac{4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac{24965 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{4536}\\ &=\frac{24965 \sqrt{1-2 x}}{15876}-\frac{(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac{31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac{4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0240554, size = 54, normalized size = 0.45 \[ \frac{(1-2 x)^{7/2} \left (2401 (279 x+179)-39944 (3 x+2)^4 \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{4235364 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*(2401*(179 + 279*x) - 39944*(2 + 3*x)^4*Hypergeometric2F1[3, 7/2, 9/2, 3/7 - (6*x)/7]))/(4235
364*(2 + 3*x)^4)

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Maple [A]  time = 0.011, size = 75, normalized size = 0.6 \begin{align*}{\frac{200}{243}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{47185}{672} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{129289}{288} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{824705}{864} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1749055}{2592}\sqrt{1-2\,x}} \right ) }-{\frac{24965\,\sqrt{21}}{47628}{\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((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^5,x)

[Out]

200/243*(1-2*x)^(1/2)+8/3*(-47185/672*(1-2*x)^(7/2)+129289/288*(1-2*x)^(5/2)-824705/864*(1-2*x)^(3/2)+1749055/
2592*(1-2*x)^(1/2))/(-6*x-4)^4-24965/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.19121, size = 161, normalized size = 1.33 \begin{align*} \frac{24965}{95256} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{200}{243} \, \sqrt{-2 \, x + 1} - \frac{1273995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 8145207 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12243385 \, \sqrt{-2 \, x + 1}}{6804 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

24965/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 200/243*sqrt(-2*x + 1
) - 1/6804*(1273995*(-2*x + 1)^(7/2) - 8145207*(-2*x + 1)^(5/2) + 17318805*(-2*x + 1)^(3/2) - 12243385*sqrt(-2
*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.26281, size = 324, normalized size = 2.68 \begin{align*} \frac{24965 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (302400 \, x^{4} + 1231065 \, x^{3} + 1526937 \, x^{2} + 762598 \, x + 134558\right )} \sqrt{-2 \, x + 1}}{95256 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/95256*(24965*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x
+ 2)) + 21*(302400*x^4 + 1231065*x^3 + 1526937*x^2 + 762598*x + 134558)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 21
6*x^2 + 96*x + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.34163, size = 147, normalized size = 1.21 \begin{align*} \frac{24965}{95256} \, \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{200}{243} \, \sqrt{-2 \, x + 1} + \frac{1273995 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 8145207 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12243385 \, \sqrt{-2 \, x + 1}}{108864 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

24965/95256*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 200/243*sqrt
(-2*x + 1) + 1/108864*(1273995*(2*x - 1)^3*sqrt(-2*x + 1) + 8145207*(2*x - 1)^2*sqrt(-2*x + 1) - 17318805*(-2*
x + 1)^(3/2) + 12243385*sqrt(-2*x + 1))/(3*x + 2)^4