3.2297 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1260 (3 x+2)^5}+\frac{106751933 \sqrt{1-2 x} \sqrt{5 x+3}}{99574272 (3 x+2)}+\frac{1057139 \sqrt{1-2 x} \sqrt{5 x+3}}{7112448 (3 x+2)^2}+\frac{47279 \sqrt{1-2 x} \sqrt{5 x+3}}{1270080 (3 x+2)^3}-\frac{6533 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{15036307 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

[Out]

(-6533*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(211680*(2 + 3*x)^4) + (47279*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1270080*(2 + 3
*x)^3) + (1057139*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7112448*(2 + 3*x)^2) + (106751933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(99574272*(2 + 3*x)) - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1260*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(18*(2 + 3*x)^6) - (15036307*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1229312*Sqrt[7])

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Rubi [A]  time = 0.0770332, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1260 (3 x+2)^5}+\frac{106751933 \sqrt{1-2 x} \sqrt{5 x+3}}{99574272 (3 x+2)}+\frac{1057139 \sqrt{1-2 x} \sqrt{5 x+3}}{7112448 (3 x+2)^2}+\frac{47279 \sqrt{1-2 x} \sqrt{5 x+3}}{1270080 (3 x+2)^3}-\frac{6533 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{15036307 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6533*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(211680*(2 + 3*x)^4) + (47279*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1270080*(2 + 3
*x)^3) + (1057139*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7112448*(2 + 3*x)^2) + (106751933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(99574272*(2 + 3*x)) - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1260*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(18*(2 + 3*x)^6) - (15036307*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1229312*Sqrt[7])

Rule 97

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

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{\int \frac{\left (-\frac{387}{4}-2595 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx}{1890}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{\int \frac{-\frac{822687}{8}-\frac{432615 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{158760}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{\int \frac{\frac{10523625}{16}-\frac{4964295 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{3333960}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}+\frac{1057139 \sqrt{1-2 x} \sqrt{3+5 x}}{7112448 (2+3 x)^2}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{\int \frac{\frac{2256323055}{32}-\frac{554997975 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{46675440}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}+\frac{1057139 \sqrt{1-2 x} \sqrt{3+5 x}}{7112448 (2+3 x)^2}+\frac{106751933 \sqrt{1-2 x} \sqrt{3+5 x}}{99574272 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{\int \frac{127883791035}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{326728080}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}+\frac{1057139 \sqrt{1-2 x} \sqrt{3+5 x}}{7112448 (2+3 x)^2}+\frac{106751933 \sqrt{1-2 x} \sqrt{3+5 x}}{99574272 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{15036307 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2458624}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}+\frac{1057139 \sqrt{1-2 x} \sqrt{3+5 x}}{7112448 (2+3 x)^2}+\frac{106751933 \sqrt{1-2 x} \sqrt{3+5 x}}{99574272 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac{15036307 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1229312}\\ &=-\frac{6533 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{47279 \sqrt{1-2 x} \sqrt{3+5 x}}{1270080 (2+3 x)^3}+\frac{1057139 \sqrt{1-2 x} \sqrt{3+5 x}}{7112448 (2+3 x)^2}+\frac{106751933 \sqrt{1-2 x} \sqrt{3+5 x}}{99574272 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}-\frac{15036307 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1229312 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.14047, size = 140, normalized size = 0.67 \[ \frac{1}{42} \left (\frac{1027 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{3073280}+\frac{579 (1-2 x)^{3/2} (5 x+3)^{7/2}}{70 (3 x+2)^5}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{(3 x+2)^6}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(2 + 3*x)^6 + (579*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(70*(2 + 3*x)^5) + (1
027*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(105552 + 654436*x + 1285720*x^2 + 814395*x^3))/(2 + 3*x)^4 - 219615*Sqrt[
7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/3073280)/42

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Maple [B]  time = 0.012, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{258155520\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 164422017045\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\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]\,{x}^{5}\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}^{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+207465599936\,{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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^7,x)

[Out]

1/258155520*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(164422017045*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^6+657688068180*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+1096146780300*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+67253717790*x^5*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+227287047960*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+307452849312*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+207465599936*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)^6

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Maxima [A]  time = 2.6667, size = 329, normalized size = 1.57 \begin{align*} \frac{15036307}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{621335}{921984} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{126 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2940 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{547 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{23520 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{31055 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{197568 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{372801 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{4597879 \, \sqrt{-10 \, x^{2} - x + 3}}{3687936 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15036307/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 621335/921984*sqrt(-10*x^2 - x +
3) + 1/126*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) - 169/2
940*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 547/23520*(-10*x^2 - x + 3
)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 31055/197568*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*
x + 8) + 372801/614656*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 4597879/3687936*sqrt(-10*x^2 - x + 3)/(3*x
+ 2)

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Fricas [A]  time = 1.58558, size = 512, normalized size = 2.45 \begin{align*} -\frac{225544605 \, \sqrt{7}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (4803836985 \, x^{5} + 16234789140 \, x^{4} + 21960917808 \, x^{3} + 14818971424 \, x^{2} + 4978384240 \, x + 665270208\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{258155520 \,{\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)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/258155520*(225544605*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14
*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(4803836985*x^5 + 16234789140*x^4 + 2
1960917808*x^3 + 14818971424*x^2 + 4978384240*x + 665270208)*sqrt(5*x + 3)*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)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 4.25718, size = 676, normalized size = 3.23 \begin{align*} \frac{15036307}{172103680} \, \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{14641 \,{\left (3081 \, \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 )}^{11} + 4888520 \, \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} + 3188465280 \, \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} - 599903001600 \, \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} - 103716175360000 \, \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} - 5302514380800000 \, \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 )}}{1843968 \,{\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 )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15036307/172103680*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)))) - 14641/1843968*(3081*sqrt(10)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 + 4888520*sq
rt(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(2
2)))^9 + 3188465280*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^7 - 599903001600*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 103716175360000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 5302514380800000*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)^6