3.1814 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac{(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac{1613 \sqrt{1-2 x}}{1037232 (3 x+2)}+\frac{1613 \sqrt{1-2 x}}{444528 (3 x+2)^2}+\frac{1613 \sqrt{1-2 x}}{158760 (3 x+2)^3}-\frac{1613 \sqrt{1-2 x}}{7560 (3 x+2)^4}+\frac{1613 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{518616 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(3/2)/(378*(2 + 3*x)^6) + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) - (1613*Sqrt[1 - 2*x])/(7560*(2
+ 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt
[1 - 2*x])/(1037232*(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*Sqrt[21])

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Rubi [A]  time = 0.0512634, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac{(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac{1613 \sqrt{1-2 x}}{1037232 (3 x+2)}+\frac{1613 \sqrt{1-2 x}}{444528 (3 x+2)^2}+\frac{1613 \sqrt{1-2 x}}{158760 (3 x+2)^3}-\frac{1613 \sqrt{1-2 x}}{7560 (3 x+2)^4}+\frac{1613 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{518616 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-(1 - 2*x)^(3/2)/(378*(2 + 3*x)^6) + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) - (1613*Sqrt[1 - 2*x])/(7560*(2
+ 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt
[1 - 2*x])/(1037232*(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{1}{378} \int \frac{\sqrt{1-2 x} (1689+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}+\frac{1613}{630} \int \frac{\sqrt{1-2 x}}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}-\frac{1613 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx}{7560}\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}+\frac{1613 \sqrt{1-2 x}}{158760 (2+3 x)^3}-\frac{1613 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{31752}\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}+\frac{1613 \sqrt{1-2 x}}{158760 (2+3 x)^3}+\frac{1613 \sqrt{1-2 x}}{444528 (2+3 x)^2}-\frac{1613 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{148176}\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}+\frac{1613 \sqrt{1-2 x}}{158760 (2+3 x)^3}+\frac{1613 \sqrt{1-2 x}}{444528 (2+3 x)^2}+\frac{1613 \sqrt{1-2 x}}{1037232 (2+3 x)}-\frac{1613 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1037232}\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}+\frac{1613 \sqrt{1-2 x}}{158760 (2+3 x)^3}+\frac{1613 \sqrt{1-2 x}}{444528 (2+3 x)^2}+\frac{1613 \sqrt{1-2 x}}{1037232 (2+3 x)}+\frac{1613 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1037232}\\ &=-\frac{(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac{137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac{1613 \sqrt{1-2 x}}{7560 (2+3 x)^4}+\frac{1613 \sqrt{1-2 x}}{158760 (2+3 x)^3}+\frac{1613 \sqrt{1-2 x}}{444528 (2+3 x)^2}+\frac{1613 \sqrt{1-2 x}}{1037232 (2+3 x)}+\frac{1613 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{518616 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0250103, size = 47, normalized size = 0.32 \[ \frac{(1-2 x)^{3/2} \left (\frac{2401 (1233 x+787)}{(3 x+2)^6}-51616 \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{31765230} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

((1 - 2*x)^(3/2)*((2401*(787 + 1233*x))/(2 + 3*x)^6 - 51616*Hypergeometric2F1[3/2, 5, 5/2, 3/7 - (6*x)/7]))/31
765230

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Maple [A]  time = 0.012, size = 84, normalized size = 0.6 \begin{align*} 23328\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ( -{\frac{1613\, \left ( 1-2\,x \right ) ^{11/2}}{49787136}}+{\frac{27421\, \left ( 1-2\,x \right ) ^{9/2}}{64012032}}-{\frac{17743\, \left ( 1-2\,x \right ) ^{7/2}}{7620480}}+{\frac{4213\, \left ( 1-2\,x \right ) ^{5/2}}{846720}}-{\frac{86837\, \left ( 1-2\,x \right ) ^{3/2}}{35271936}}-{\frac{11291\,\sqrt{1-2\,x}}{5038848}} \right ) }+{\frac{1613\,\sqrt{21}}{10890936}{\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)^2*(1-2*x)^(1/2)/(2+3*x)^7,x)

[Out]

23328*(-1613/49787136*(1-2*x)^(11/2)+27421/64012032*(1-2*x)^(9/2)-17743/7620480*(1-2*x)^(7/2)+4213/846720*(1-2
*x)^(5/2)-86837/35271936*(1-2*x)^(3/2)-11291/5038848*(1-2*x)^(1/2))/(-6*x-4)^6+1613/10890936*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.66551, size = 197, normalized size = 1.33 \begin{align*} -\frac{1613}{21781872} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1959795 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 25912845 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 140843934 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 300985146 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 148925455 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 135548455 \, \sqrt{-2 \, x + 1}}{2593080 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1613/21781872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/2593080*(1959795
*(-2*x + 1)^(11/2) - 25912845*(-2*x + 1)^(9/2) + 140843934*(-2*x + 1)^(7/2) - 300985146*(-2*x + 1)^(5/2) + 148
925455*(-2*x + 1)^(3/2) + 135548455*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 +
185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 1.6454, size = 423, normalized size = 2.86 \begin{align*} \frac{8065 \, \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (1959795 \, x^{5} + 8056935 \, x^{4} + 14197626 \, x^{3} + 1791558 \, x^{2} - 7772840 \, x - 3136864\right )} \sqrt{-2 \, x + 1}}{108909360 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/108909360*(8065*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x - sqrt(
21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1959795*x^5 + 8056935*x^4 + 14197626*x^3 + 1791558*x^2 - 7772840*x -
3136864)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [A]  time = 2.28459, size = 178, normalized size = 1.2 \begin{align*} -\frac{1613}{21781872} \, \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{1959795 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 25912845 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 140843934 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 300985146 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 148925455 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 135548455 \, \sqrt{-2 \, x + 1}}{165957120 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1613/21781872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1659571
20*(1959795*(2*x - 1)^5*sqrt(-2*x + 1) + 25912845*(2*x - 1)^4*sqrt(-2*x + 1) + 140843934*(2*x - 1)^3*sqrt(-2*x
+ 1) + 300985146*(2*x - 1)^2*sqrt(-2*x + 1) - 148925455*(-2*x + 1)^(3/2) - 135548455*sqrt(-2*x + 1))/(3*x + 2
)^6