\(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx\) [2335]

Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac {240911 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \]

[Out]

3/28*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4+181/168*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-240911/21952*arctan(1
/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1991/224*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-21901/3136*(1-2
*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {240911 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}}+\frac {3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac {1991 (5 x+3)^{3/2} \sqrt {1-2 x}}{224 (3 x+2)^2}-\frac {21901 \sqrt {5 x+3} \sqrt {1-2 x}}{3136 (3 x+2)} \]

[In]

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

[Out]

(-21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) + (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(28*(2 + 3*x)^4) +
(181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(168*(2 + 3*x)^3) + (1991*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(224*(2 + 3*x)
^2) - (240911*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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*} \text {integral}& = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181}{56} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx \\ & = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991}{112} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx \\ & = \frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {21901}{448} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272} \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136} \\ & = -\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac {240911 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (541680+2381420 x+3485960 x^2+1705089 x^3\right )}{(2+3 x)^4}-722733 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{65856} \]

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(541680 + 2381420*x + 3485960*x^2 + 1705089*x^3))/(2 + 3*x)^4 - 722733*Sqrt[7]
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/65856

Maple [A] (verified)

Time = 3.72 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1705089 x^{3}+3485960 x^{2}+2381420 x +541680\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{9408 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {240911 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43904 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(129\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (58541373 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}+7583520 \sqrt {-10 x^{2}-x +3}\right )}{131712 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(250\)

[In]

int((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^5,x,method=_RETURNVERBOSE)

[Out]

-1/9408*(-1+2*x)*(3+5*x)^(1/2)*(1705089*x^3+3485960*x^2+2381420*x+541680)/(2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*
((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+240911/43904*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67
+111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=-\frac {722733 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1705089 \, x^{3} + 3485960 \, x^{2} + 2381420 \, x + 541680\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{131712 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

[In]

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

[Out]

-1/131712*(722733*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1705089*x^3 + 3485960*x^2 + 2381420*x + 541680)*sqrt(5*x + 3)*sqrt(-
2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

Sympy [F]

\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{5}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {240911}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {9955}{2352} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{168 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {5973 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {73667 \, \sqrt {-10 \, x^{2} - x + 3}}{9408 \, {\left (3 \, x + 2\right )}} \]

[In]

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

[Out]

240911/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 9955/2352*sqrt(-10*x^2 - x + 3) + 1/4
*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 169/168*(-10*x^2 - x + 3)^(3/2)/(27*x^3 +
54*x^2 + 36*x + 8) + 5973/1568*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 73667/9408*sqrt(-10*x^2 - x + 3)/(
3*x + 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).

Time = 0.51 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\frac {240911}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (543 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 696920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 156094400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {11919936000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {47679744000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4704 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

[In]

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

[Out]

240911/439040*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)))) - 1331/4704*sqrt(10)*(543*((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 - 696920*((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 - 156094400*((
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 -
11919936000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 47679744000*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)^4

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^5} \,d x \]

[In]

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

[Out]

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