3.1941 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac{41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{205 \sqrt{1-2 x}}{444528 (3 x+2)}+\frac{205 \sqrt{1-2 x}}{190512 (3 x+2)^2}-\frac{205 \sqrt{1-2 x}}{13608 (3 x+2)^3}+\frac{205 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

[Out]

(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) - (41*(1 - 2*x)^(5/2))/(378*(2 + 3*x)^5) + (205*(1 - 2*x)^(3/2))/(4536*(2 +
3*x)^4) - (205*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3) + (205*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (205*Sqrt[1 - 2
*x])/(444528*(2 + 3*x)) + (205*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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Rubi [A]  time = 0.0453164, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac{(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac{41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{205 \sqrt{1-2 x}}{444528 (3 x+2)}+\frac{205 \sqrt{1-2 x}}{190512 (3 x+2)^2}-\frac{205 \sqrt{1-2 x}}{13608 (3 x+2)^3}+\frac{205 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) - (41*(1 - 2*x)^(5/2))/(378*(2 + 3*x)^5) + (205*(1 - 2*x)^(3/2))/(4536*(2 +
3*x)^4) - (205*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3) + (205*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (205*Sqrt[1 - 2
*x])/(444528*(2 + 3*x)) + (205*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}+\frac{205}{126} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}-\frac{205}{378} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{205 \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx}{1512}\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{205 \sqrt{1-2 x}}{13608 (2+3 x)^3}-\frac{205 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{13608}\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{205 \sqrt{1-2 x}}{13608 (2+3 x)^3}+\frac{205 \sqrt{1-2 x}}{190512 (2+3 x)^2}-\frac{205 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{205 \sqrt{1-2 x}}{13608 (2+3 x)^3}+\frac{205 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{205 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{205 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{444528}\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{205 \sqrt{1-2 x}}{13608 (2+3 x)^3}+\frac{205 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{205 \sqrt{1-2 x}}{444528 (2+3 x)}+\frac{205 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{444528}\\ &=\frac{(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac{41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{205 \sqrt{1-2 x}}{13608 (2+3 x)^3}+\frac{205 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{205 \sqrt{1-2 x}}{444528 (2+3 x)}+\frac{205 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0178774, size = 42, normalized size = 0.28 \[ \frac{(1-2 x)^{7/2} \left (\frac{823543}{(3 x+2)^6}-13120 \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{103766418} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*(823543/(2 + 3*x)^6 - 13120*Hypergeometric2F1[7/2, 6, 9/2, 3/7 - (6*x)/7]))/103766418

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Maple [A]  time = 0.01, size = 84, normalized size = 0.6 \begin{align*} -46656\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ({\frac{205\, \left ( 1-2\,x \right ) ^{11/2}}{42674688}}-{\frac{3485\, \left ( 1-2\,x \right ) ^{9/2}}{54867456}}-{\frac{439\, \left ( 1-2\,x \right ) ^{7/2}}{3919104}}+{\frac{451\, \left ( 1-2\,x \right ) ^{5/2}}{559872}}-{\frac{24395\, \left ( 1-2\,x \right ) ^{3/2}}{30233088}}+{\frac{10045\,\sqrt{1-2\,x}}{30233088}} \right ) }+{\frac{205\,\sqrt{21}}{4667544}{\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+3*x)^7,x)

[Out]

-46656*(205/42674688*(1-2*x)^(11/2)-3485/54867456*(1-2*x)^(9/2)-439/3919104*(1-2*x)^(7/2)+451/559872*(1-2*x)^(
5/2)-24395/30233088*(1-2*x)^(3/2)+10045/30233088*(1-2*x)^(1/2))/(-6*x-4)^6+205/4667544*arctanh(1/7*21^(1/2)*(1
-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.23539, size = 197, normalized size = 1.33 \begin{align*} -\frac{205}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49815 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 658665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 1161594 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 8353422 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 8367485 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 3445435 \, \sqrt{-2 \, x + 1}}{222264 \,{\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((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-205/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/222264*(49815*(-2*
x + 1)^(11/2) - 658665*(-2*x + 1)^(9/2) - 1161594*(-2*x + 1)^(7/2) + 8353422*(-2*x + 1)^(5/2) - 8367485*(-2*x
+ 1)^(3/2) + 3445435*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 = 1.53072, size = 406, normalized size = 2.74 \begin{align*} \frac{205 \, \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 (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt{-2 \, x + 1}}{9335088 \,{\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((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/9335088*(205*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*(49815*x^5 + 204795*x^4 - 824526*x^3 - 176850*x^2 + 154312*x - 51904)*sqr
t(-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((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.94455, size = 178, normalized size = 1.2 \begin{align*} -\frac{205}{9335088} \, \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{49815 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 658665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 1161594 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 8353422 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 8367485 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3445435 \, \sqrt{-2 \, x + 1}}{14224896 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-205/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/14224896*
(49815*(2*x - 1)^5*sqrt(-2*x + 1) + 658665*(2*x - 1)^4*sqrt(-2*x + 1) - 1161594*(2*x - 1)^3*sqrt(-2*x + 1) - 8
353422*(2*x - 1)^2*sqrt(-2*x + 1) + 8367485*(-2*x + 1)^(3/2) - 3445435*sqrt(-2*x + 1))/(3*x + 2)^6