3.2272 \(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{840 (3 x+2)^4}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]

[Out]

-5591773/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/15*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*
x)^5+37/840*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+403/1680*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+14023/9408*(1
-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1466281/131712*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.07, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{840 (3 x+2)^4}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(840*(2 + 3*x)^4) + (403*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (14023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (1466
281*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (5591773*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(43904*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 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 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 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)^6} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {1}{420} \int \frac {\frac {1341}{4}-555 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {\int \frac {\frac {265125}{8}-42315 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{8820}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {\int \frac {\frac {31687635}{16}-\frac {7362075 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{123480}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {\int \frac {1761408495}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{864360}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {5591773 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {5591773 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 135, normalized size = 0.75 \[ \frac {1}{35} \left (\frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (4398843 x^2+6119462 x+2067760\right )}{(3 x+2)^3}-5591773 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{43904}+\frac {111 (1-2 x)^{3/2} (5 x+3)^{3/2}}{8 (3 x+2)^4}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5 + (111*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(8*(2 + 3*x)^4) + (5*
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2067760 + 6119462*x + 4398843*x^2))/(2 + 3*x)^3 - 5591773*Sqrt[7]*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/43904)/35

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fricas [A]  time = 1.01, size = 131, normalized size = 0.73 \[ -\frac {27958865 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (197947935 \, x^{4} + 536695650 \, x^{3} + 546004068 \, x^{2} + 247045192 \, x + 41933792\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

-1/3073280*(27958865*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*(197947935*x^4 + 536695650*x^3 + 546004068*x^2 + 2470
45192*x + 41933792)*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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giac [B]  time = 2.77, size = 426, normalized size = 2.37 \[ \frac {5591773}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (46213 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 85961680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 30665564160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 4732042112000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {284050977280000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1136203909120000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

5591773/6146560*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 121/21952*sqrt(10)*(46213*((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 - 85961680*((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 - 306655
64160*((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 - 4732042112000*((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 - 284050977280000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1136203909120000*
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*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5

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maple [B]  time = 0.02, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (6794004195 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3073280*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(6794004195*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^5+22646680650*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^3*arctan(
1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+20130382800*7^(1/2)*x^2*arctan(
1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+6710127600*7^(1/2)*x*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+894683680*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+587073088*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+
3)^(1/2)/(3*x+2)^5

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maxima [A]  time = 1.30, size = 198, normalized size = 1.10 \[ \frac {5591773}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {231065}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1305 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {138639 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1709881 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]

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)^6,x, algorithm="maxima")

[Out]

5591773/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 231065/32928*sqrt(-10*x^2 - x + 3)
+ 3/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 111/280*(-10*x^2 - x +
3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1305/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x +
8) + 138639/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1709881/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 20.39, size = 1745, normalized size = 9.69 \[ \text {result too large to display} \]

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)^6,x)

[Out]

((45503192879*((1 - 2*x)^(1/2) - 1)^7)/(478515625*(3^(1/2) - (5*x + 3)^(1/2))^7) - (2981176*((1 - 2*x)^(1/2) -
1)^3)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (61599781879*((1 - 2*x)^(1/2) - 1)^5)/(3349609375*(3^(1/2) -
(5*x + 3)^(1/2))^5) - (22334164*((1 - 2*x)^(1/2) - 1))/(3349609375*(3^(1/2) - (5*x + 3)^(1/2))) - (13758944676
7*((1 - 2*x)^(1/2) - 1)^9)/(535937500*(3^(1/2) - (5*x + 3)^(1/2))^9) + (137589446767*((1 - 2*x)^(1/2) - 1)^11)
/(214375000*(3^(1/2) - (5*x + 3)^(1/2))^11) - (45503192879*((1 - 2*x)^(1/2) - 1)^13)/(30625000*(3^(1/2) - (5*x
+ 3)^(1/2))^13) + (61599781879*((1 - 2*x)^(1/2) - 1)^15)/(34300000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (372647*
((1 - 2*x)^(1/2) - 1)^17)/(400*(3^(1/2) - (5*x + 3)^(1/2))^17) + (5583541*((1 - 2*x)^(1/2) - 1)^19)/(219520*(3
^(1/2) - (5*x + 3)^(1/2))^19) + (12070766*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(133984375*(3^(1/2) - (5*x + 3)^(1/
2))^2) + (2979759193*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(669921875*(3^(1/2) - (5*x + 3)^(1/2))^4) + (132301459*3
^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(957031250*(3^(1/2) - (5*x + 3)^(1/2))^6) - (442781811679*3^(1/2)*((1 - 2*x)^(
1/2) - 1)^8)/(6699218750*(3^(1/2) - (5*x + 3)^(1/2))^8) + (1165566494503*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(47
85156250*(3^(1/2) - (5*x + 3)^(1/2))^10) - (442781811679*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(1071875000*(3^(1/2
) - (5*x + 3)^(1/2))^12) + (132301459*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(24500000*(3^(1/2) - (5*x + 3)^(1/2))^
14) + (2979759193*3^(1/2)*((1 - 2*x)^(1/2) - 1)^16)/(2744000*(3^(1/2) - (5*x + 3)^(1/2))^16) + (6035383*3^(1/2
)*((1 - 2*x)^(1/2) - 1)^18)/(43904*(3^(1/2) - (5*x + 3)^(1/2))^18))/((11776*((1 - 2*x)^(1/2) - 1)^2)/(390625*(
3^(1/2) - (5*x + 3)^(1/2))^2) + (1082112*((1 - 2*x)^(1/2) - 1)^4)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^4) - (4
163328*((1 - 2*x)^(1/2) - 1)^6)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (1029696*((1 - 2*x)^(1/2) - 1)^8)/(3
90625*(3^(1/2) - (5*x + 3)^(1/2))^8) - (15348576*((1 - 2*x)^(1/2) - 1)^10)/(9765625*(3^(1/2) - (5*x + 3)^(1/2)
)^10) + (257424*((1 - 2*x)^(1/2) - 1)^12)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^12) - (260208*((1 - 2*x)^(1/2) -
1)^14)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^14) + (16908*((1 - 2*x)^(1/2) - 1)^16)/(125*(3^(1/2) - (5*x + 3)^(1/2
))^16) + (46*((1 - 2*x)^(1/2) - 1)^18)/(3^(1/2) - (5*x + 3)^(1/2))^18 + ((1 - 2*x)^(1/2) - 1)^20/(3^(1/2) - (5
*x + 3)^(1/2))^20 - (201216*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (280012
8*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^5) + (6514944*3^(1/2)*((1 - 2*x)^(1/2)
- 1)^7)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^7) - (24309312*3^(1/2)*((1 - 2*x)^(1/2) - 1)^9)/(1953125*(3^(1/2
) - (5*x + 3)^(1/2))^9) + (12154656*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^11)
- (814368*3^(1/2)*((1 - 2*x)^(1/2) - 1)^13)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^13) + (87504*3^(1/2)*((1 - 2*x)
^(1/2) - 1)^15)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^15) + (1572*3^(1/2)*((1 - 2*x)^(1/2) - 1)^17)/(25*(3^(1/2) -
(5*x + 3)^(1/2))^17) + (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^19)/(3^(1/2) - (5*x + 3)^(1/2))^19 - (3072*3^(1/2)*((
1 - 2*x)^(1/2) - 1))/(1953125*(3^(1/2) - (5*x + 3)^(1/2))) + 1024/9765625) - (5591773*7^(1/2)*atan(((5591773*7
^(1/2)*((16775319*3^(1/2))/686000 + (16775319*((1 - 2*x)^(1/2) - 1))/(1372000*(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)*5591773i)/614656 - (16775319*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(
274400*(3^(1/2) - (5*x + 3)^(1/2))^2)))/614656 + (5591773*7^(1/2)*((16775319*3^(1/2))/686000 + (16775319*((1 -
2*x)^(1/2) - 1))/(1372000*(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)*5591
773i)/614656 - (16775319*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(274400*(3^(1/2) - (5*x + 3)^(1/2))^2)))/614656)/((7
^(1/2)*((16775319*3^(1/2))/686000 + (16775319*((1 - 2*x)^(1/2) - 1))/(1372000*(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)*5591773i)/614656 - (16775319*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(
274400*(3^(1/2) - (5*x + 3)^(1/2))^2))*5591773i)/614656 - (7^(1/2)*((16775319*3^(1/2))/686000 + (16775319*((1
- 2*x)^(1/2) - 1))/(1372000*(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)*559
1773i)/614656 - (16775319*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(274400*(3^(1/2) - (5*x + 3)^(1/2))^2))*5591773i)/6
14656 + (31267925283529*((1 - 2*x)^(1/2) - 1)^2)/(24094515200*(3^(1/2) - (5*x + 3)^(1/2))^2) + 31267925283529/
60236288000)))/307328

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \]

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)**6,x)

[Out]

Integral(sqrt(1 - 2*x)*sqrt(5*x + 3)/(3*x + 2)**6, x)

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