3.2358 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{33 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(3136*(2 + 3*x)^
2) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(560*(2 + 3*x)^3) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(5*(2 + 3*x)^5)
+ (33*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(43904*Sqrt[7])

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Rubi [A]  time = 0.053783, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{33 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6,x]

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(3136*(2 + 3*x)^
2) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(560*(2 + 3*x)^3) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(5*(2 + 3*x)^5)
+ (33*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(43904*Sqrt[7])

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33}{10} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{363}{80} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{1331}{224} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{43923 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{6272}\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{483153 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{483153 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac{(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac{33 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0675232, size = 84, normalized size = 0.47 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (15899035 x^4+46076650 x^3+47906548 x^2+21437032 x+3507552\right )}{(3 x+2)^5}-2415765 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1536640} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6,x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3507552 + 21437032*x + 47906548*x^2 + 46076650*x^3 + 15899035*x^4))/(2 + 3*x)
^5 - 2415765*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1536640

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Maple [B]  time = 0.012, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 587030895\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+670691672\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^6,x)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(587030895*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^
5+1956769650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2609026200*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+222586490*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+645073100*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+670691672*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+
3*x)^5

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Maxima [A]  time = 1.80861, size = 306, normalized size = 1.7 \begin{align*} \frac{90695}{230496} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1221 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{54417 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{153664 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{738705}{153664} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{483153}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{650859}{307328} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{215303 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{921984 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

90695/230496*(-10*x^2 - x + 3)^(3/2) - 1/35*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 +
240*x + 32) + 33/392*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1221/5488*(-10*x^2 - x
+ 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 54417/153664*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 738705/15
3664*sqrt(-10*x^2 - x + 3)*x + 483153/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 65085
9/307328*sqrt(-10*x^2 - x + 3) + 215303/921984*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.60429, size = 431, normalized size = 2.39 \begin{align*} -\frac{2415765 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (15899035 \, x^{4} + 46076650 \, x^{3} + 47906548 \, x^{2} + 21437032 \, x + 3507552\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/3073280*(2415765*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x +
20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(15899035*x^4 + 46076650*x^3 + 47906548*x^2 + 21437032
*x + 3507552)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 4.04311, size = 594, normalized size = 3.3 \begin{align*} \frac{483153}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{161051 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2007040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 307328000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 18439680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{21952 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

483153/6146560*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 161051/21952*(3*sqrt(10)*((sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 3920*sqrt(10)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 200
7040*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^5 - 307328000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 18439680000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)
- 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)
- 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5