3.23.73 \(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx\) [2273]

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

[Out]

-588912203/8605184*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/18*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2
+3*x)^6+37/1260*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5+10921/70560*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+126799
/141120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+4429459/790272*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+463266973/1
1063808*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 209, normalized size of antiderivative = 1.00, 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 = {99, 156, 12, 95, 210} \begin {gather*} -\frac {588912203 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1229312 \sqrt {7}}+\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(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 12

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

Rule 95

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 99

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 156

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] && ILtQ[m, -1]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 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 \text {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 = 0.40, size = 91, normalized size = 0.44 \begin {gather*} \frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (8835086144+65287037520 x+193055073632 x^2+285550790544 x^3+211260697020 x^4+62541041355 x^5\right )}{121 (2+3 x)^6}-24335215 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{43025920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(121*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8835086144 + 65287037520*x + 193055073632*x^2 + 285550790544*x^3 + 21126
0697020*x^4 + 62541041355*x^5))/(121*(2 + 3*x)^6) - 24335215*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])]))/43025920

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(164)=328\).
time = 0.14, size = 346, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (62541041355 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6146560 \left (2+3 x \right )^{6} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {588912203 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2146584979935 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{86051840 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

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,method=_RETURNVERBOSE)

[Out]

1/86051840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2146584979935*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/
2)*x^6+8586339919740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(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

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 244, normalized size = 1.17 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.75, size = 146, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{7}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
time = 0.86, size = 484, normalized size = 2.32 \begin {gather*} \frac {588912203}{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 {121 \, \sqrt {10} {\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 )}}{614656 \, {\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 {gather*}

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]

588912203/172103680*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)))) - 121/614656*sqrt(10)*(4867043*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 - 127661584
40*((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)/(s
qrt(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) - sq
rt(22))/sqrt(5*x + 3) + 33505440440729600000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sq
rt(-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

________________________________________________________________________________________

Mupad [B]
time = 26.48, size = 2500, normalized size = 11.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((16944332789147*((1 - 2*x)^(1/2) - 1)^7)/(234472656250*(3^(1/2) - (5*x + 3)^(1/2))^7) - (166614000612*((1 - 2
*x)^(1/2) - 1)^3)/(117236328125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (3113868401706*((1 - 2*x)^(1/2) - 1)^5)/(8374
0234375*(3^(1/2) - (5*x + 3)^(1/2))^5) - (471037564*((1 - 2*x)^(1/2) - 1))/(117236328125*(3^(1/2) - (5*x + 3)^
(1/2))) + (788747583108129*((1 - 2*x)^(1/2) - 1)^9)/(2344726562500*(3^(1/2) - (5*x + 3)^(1/2))^9) - (235760192
51663873*((1 - 2*x)^(1/2) - 1)^11)/(11723632812500*(3^(1/2) - (5*x + 3)^(1/2))^11) + (23576019251663873*((1 -
2*x)^(1/2) - 1)^13)/(4689453125000*(3^(1/2) - (5*x + 3)^(1/2))^13) - (788747583108129*((1 - 2*x)^(1/2) - 1)^15
)/(150062500000*(3^(1/2) - (5*x + 3)^(1/2))^15) - (16944332789147*((1 - 2*x)^(1/2) - 1)^17)/(2401000000*(3^(1/
2) - (5*x + 3)^(1/2))^17) + (1556934200853*((1 - 2*x)^(1/2) - 1)^19)/(68600000*(3^(1/2) - (5*x + 3)^(1/2))^19)
+ (41653500153*((1 - 2*x)^(1/2) - 1)^21)/(7683200*(3^(1/2) - (5*x + 3)^(1/2))^21) + (117759391*((1 - 2*x)^(1/
2) - 1)^23)/(1229312*(3^(1/2) - (5*x + 3)^(1/2))^23) + (38864901622*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(58618164
0625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (655789856256*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(117236328125*(3^(1/2) -
(5*x + 3)^(1/2))^4) + (8465281916419*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(234472656250*(3^(1/2) - (5*x + 3)^(1/2)
)^6) - (162652045859857*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(586181640625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (21120
963153193039*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(23447265625000*(3^(1/2) - (5*x + 3)^(1/2))^10) - (385565113685
6339*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(1674804687500*(3^(1/2) - (5*x + 3)^(1/2))^12) + (21120963153193039*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^14)/(3751562500000*(3^(1/2) - (5*x + 3)^(1/2))^14) - (162652045859857*3^(1/2)*((1 -
2*x)^(1/2) - 1)^16)/(15006250000*(3^(1/2) - (5*x + 3)^(1/2))^16) + (8465281916419*3^(1/2)*((1 - 2*x)^(1/2) -
1)^18)/(960400000*(3^(1/2) - (5*x + 3)^(1/2))^18) + (2561679126*3^(1/2)*((1 - 2*x)^(1/2) - 1)^20)/(300125*(3^(
1/2) - (5*x + 3)^(1/2))^20) + (19432450811*3^(1/2)*((1 - 2*x)^(1/2) - 1)^22)/(30732800*(3^(1/2) - (5*x + 3)^(1
/2))^22))/((1744896*((1 - 2*x)^(1/2) - 1)^2)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (12125184*((1 - 2*x)^
(1/2) - 1)^4)/(48828125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (10098688*((1 - 2*x)^(1/2) - 1)^6)/(244140625*(3^(1/2
) - (5*x + 3)^(1/2))^6) - (59116032*((1 - 2*x)^(1/2) - 1)^8)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (708598
4256*((1 - 2*x)^(1/2) - 1)^10)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^10) - (20806140864*((1 - 2*x)^(1/2) - 1)
^12)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (1771496064*((1 - 2*x)^(1/2) - 1)^14)/(9765625*(3^(1/2) - (5
*x + 3)^(1/2))^14) - (3694752*((1 - 2*x)^(1/2) - 1)^16)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^16) + (157792*((1 -
2*x)^(1/2) - 1)^18)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^18) + (47364*((1 - 2*x)^(1/2) - 1)^20)/(125*(3^(1/2) -
(5*x + 3)^(1/2))^20) + (1704*((1 - 2*x)^(1/2) - 1)^22)/(25*(3^(1/2) - (5*x + 3)^(1/2))^22) + ((1 - 2*x)^(1/2)
- 1)^24/(3^(1/2) - (5*x + 3)^(1/2))^24 - (1548288*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(48828125*(3^(1/2) - (5*x
+ 3)^(1/2))^3) - (81764352*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^5) + (47806
1568*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^7) - (206710272*3^(1/2)*((1 - 2*x
)^(1/2) - 1)^9)/(48828125*(3^(1/2) - (5*x + 3)^(1/2))^9) + (1284636672*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(2441
40625*(3^(1/2) - (5*x + 3)^(1/2))^11) - (642318336*3^(1/2)*((1 - 2*x)^(1/2) - 1)^13)/(48828125*(3^(1/2) - (5*x
+ 3)^(1/2))^13) + (25838784*3^(1/2)*((1 - 2*x)^(1/2) - 1)^15)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^15) - (1493
9424*3^(1/2)*((1 - 2*x)^(1/2) - 1)^17)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^17) + (638784*3^(1/2)*((1 - 2*x)^(1/
2) - 1)^19)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^19) + (3024*3^(1/2)*((1 - 2*x)^(1/2) - 1)^21)/(25*(3^(1/2) - (5*
x + 3)^(1/2))^21) + (36*3^(1/2)*((1 - 2*x)^(1/2) - 1)^23)/(5*(3^(1/2) - (5*x + 3)^(1/2))^23) - (73728*3^(1/2)*
((1 - 2*x)^(1/2) - 1))/(244140625*(3^(1/2) - (5*x + 3)^(1/2))) + 4096/244140625) - (588912203*7^(1/2)*atan(((5
88912203*7^(1/2)*((1766736609*3^(1/2))/19208000 + (1766736609*((1 - 2*x)^(1/2) - 1))/(38416000*(3^(1/2) - (5*x
+ 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1
- 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*588912203i)/17210368 - (1766736609*3^(1/2)*((1
- 2*x)^(1/2) - 1)^2)/(7683200*(3^(1/2) - (5*x + 3)^(1/2))^2)))/17210368 + (588912203*7^(1/2)*((1766736609*3^(
1/2))/19208000 + (1766736609*((1 - 2*x)^(1/2) - 1))/(38416000*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((
1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2)
- (5*x + 3)^(1/2))) - 536/125)*588912203i)/17210368 - (1766736609*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(7683200*(3
^(1/2) - (5*x + 3)^(1/2))^2)))/17210368)/((7^(1/2)*((1766736609*3^(1/2))/19208000 + (1766736609*((1 - 2*x)^(1/
2) - 1))/(38416000*(3^(1/2) - (5*x + 3)^(1/2)))...

________________________________________________________________________________________