3.2273 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{463266973 \sqrt{1-2 x} \sqrt{5 x+3}}{11063808 (3 x+2)}+\frac{4429459 \sqrt{1-2 x} \sqrt{5 x+3}}{790272 (3 x+2)^2}+\frac{126799 \sqrt{1-2 x} \sqrt{5 x+3}}{141120 (3 x+2)^3}+\frac{10921 \sqrt{1-2 x} \sqrt{5 x+3}}{70560 (3 x+2)^4}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{1260 (3 x+2)^5}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{18 (3 x+2)^6}-\frac{588912203 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1260*(2 + 3*x)^5) + (10921
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(70560*(2 + 3*x)^4) + (126799*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(141120*(2 + 3*x)^3)
+ (4429459*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(790272*(2 + 3*x)^2) + (463266973*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(110638
08*(2 + 3*x)) - (588912203*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1229312*Sqrt[7])

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Rubi [A]  time = 0.0805943, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {97, 151, 12, 93, 204} \[ \frac{463266973 \sqrt{1-2 x} \sqrt{5 x+3}}{11063808 (3 x+2)}+\frac{4429459 \sqrt{1-2 x} \sqrt{5 x+3}}{790272 (3 x+2)^2}+\frac{126799 \sqrt{1-2 x} \sqrt{5 x+3}}{141120 (3 x+2)^3}+\frac{10921 \sqrt{1-2 x} \sqrt{5 x+3}}{70560 (3 x+2)^4}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{1260 (3 x+2)^5}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{18 (3 x+2)^6}-\frac{588912203 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1260*(2 + 3*x)^5) + (10921
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(70560*(2 + 3*x)^4) + (126799*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(141120*(2 + 3*x)^3)
+ (4429459*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(790272*(2 + 3*x)^2) + (463266973*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(110638
08*(2 + 3*x)) - (588912203*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 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} \sqrt{3+5 x}}{(2+3 x)^7} \, dx &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{-\frac{1}{2}-10 x}{\sqrt{1-2 x} (2+3 x)^6 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{1}{630} \int \frac{\frac{1667}{4}-740 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{\int \frac{\frac{450753}{8}-\frac{163815 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{17640}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{\int \frac{\frac{84023625}{16}-\frac{13313895 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{370440}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{4429459 \sqrt{1-2 x} \sqrt{3+5 x}}{790272 (2+3 x)^2}+\frac{\int \frac{\frac{10013101455}{32}-\frac{2325465975 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{5186160}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{4429459 \sqrt{1-2 x} \sqrt{3+5 x}}{790272 (2+3 x)^2}+\frac{463266973 \sqrt{1-2 x} \sqrt{3+5 x}}{11063808 (2+3 x)}+\frac{\int \frac{556522031835}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{36303120}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{4429459 \sqrt{1-2 x} \sqrt{3+5 x}}{790272 (2+3 x)^2}+\frac{463266973 \sqrt{1-2 x} \sqrt{3+5 x}}{11063808 (2+3 x)}+\frac{588912203 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2458624}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{4429459 \sqrt{1-2 x} \sqrt{3+5 x}}{790272 (2+3 x)^2}+\frac{463266973 \sqrt{1-2 x} \sqrt{3+5 x}}{11063808 (2+3 x)}+\frac{588912203 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1229312}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{1260 (2+3 x)^5}+\frac{10921 \sqrt{1-2 x} \sqrt{3+5 x}}{70560 (2+3 x)^4}+\frac{126799 \sqrt{1-2 x} \sqrt{3+5 x}}{141120 (2+3 x)^3}+\frac{4429459 \sqrt{1-2 x} \sqrt{3+5 x}}{790272 (2+3 x)^2}+\frac{463266973 \sqrt{1-2 x} \sqrt{3+5 x}}{11063808 (2+3 x)}-\frac{588912203 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1229312 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.201292, size = 193, normalized size = 0.92 \[ \frac{1}{42} \left (\frac{999 (1-2 x)^{3/2} (5 x+3)^{3/2}}{70 (3 x+2)^5}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^6}+\frac{3 \left (64324848 (1-2 x)^{3/2} (5 x+3)^{3/2}+5 (3 x+2) \left (53882360 (1-2 x)^{3/2} (5 x+3)^{3/2}+4867043 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (37 x+20)-121 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )}{3073280 (3 x+2)^4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6 + (999*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(70*(2 + 3*x)^5) + (3
*(64324848*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2) + 5*(2 + 3*x)*(53882360*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2) + 4867043*(
2 + 3*x)*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(20 + 37*x) - 121*Sqrt[7]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq
rt[3 + 5*x])]))))/(3073280*(2 + 3*x)^4))/42

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Maple [B]  time = 0.015, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{86051840\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2146584979935\,\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+2702771030848\,{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)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x)

[Out]

1/86051840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2146584979935*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^6+8586339919740*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+14310566532900*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+875574578970*x^5*(-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^3+2957649758280*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+3997711067616*x^3*(-10*x^2-x+3)^(1/2)[phone]
640*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2702771030848*x^2*(-10*x^2-x+3)^(1/2)+1884519
04960*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]
16*(-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 = 3.97903, size = 329, normalized size = 1.57 \begin{align*} \frac{588912203}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{24335215}{921984} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{333 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{980 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{11721 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{7840 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{137455 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{14601129 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{180080591 \, \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((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

588912203/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 24335215/921984*sqrt(-10*x^2 -
x + 3) + 1/14*(-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) + 333
/980*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 11721/7840*(-10*x^2 - x +
3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 137455/21952*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 3
6*x + 8) + 14601129/614656*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 180080591/3687936*sqrt(-10*x^2 - x + 3
)/(3*x + 2)

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Fricas [A]  time = 1.79858, size = 520, normalized size = 2.49 \begin{align*} -\frac{2944561015 \, \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 (62541041355 \, x^{5} + 211260697020 \, x^{4} + 285550790544 \, x^{3} + 193055073632 \, x^{2} + 65287037520 \, x + 8835086144\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{86051840 \,{\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)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/86051840*(2944561015*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*(62541041355*x^5 + 211260697020*x^4 +
285550790544*x^3 + 193055073632*x^2 + 65287037520*x + 8835086144)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 29
16*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 - 2 x} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 5.00488, size = 660, normalized size = 3.16 \begin{align*} \frac{121}{172103680} \, \sqrt{5}{\left (4867043 \, \sqrt{70} \sqrt{2}{\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{280 \, \sqrt{2}{\left (4867043 \,{\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} - 12766158440 \,{\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} - 6076175020160 \,{\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} - 1409555377484800 \,{\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} - 169516778170880000 \,{\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} - \frac{8376360110182400000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{33505440440729600000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\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}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/172103680*sqrt(5)*(4867043*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 280*sqrt(2)*(4867043*((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)))^11 - 1276615
8440*((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 - 6076175020160*((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)))^7 - 1409555377484800*((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 - 169516778170880000*((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 - 8376360110182400000*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) + 33505440440729600000*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)