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

Optimal. Leaf size=209 \[ \frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-395307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (11979*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12544*(2 + 3
*x)^2) - (1089*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2240*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 +
3*x)^6) + (9*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (297*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(160*(2 +
3*x)^4) - (4348377*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

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 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 {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9}{4} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297}{40} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {3267}{320} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {11979}{896} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {395307 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 138, normalized size = 0.66 \begin {gather*} \frac {99 \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 )}{6146560}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 + 3*x)^6) + (9*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (9
9*((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])]))/6146560

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.46, size = 154, normalized size = 0.74 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {135 (1-2 x)^5}{(5 x+3)^5}+\frac {5355 (1-2 x)^4}{(5 x+3)^4}+\frac {87318 (1-2 x)^3}{(5 x+3)^3}-\frac {510874 (1-2 x)^2}{(5 x+3)^2}-\frac {1836765 (1-2 x)}{5 x+3}-2268945\right )}{878080 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^6}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-161051*Sqrt[1 - 2*x]*(-2268945 + (135*(1 - 2*x)^5)/(3 + 5*x)^5 + (5355*(1 - 2*x)^4)/(3 + 5*x)^4 + (87318*(1
- 2*x)^3)/(3 + 5*x)^3 - (510874*(1 - 2*x)^2)/(3 + 5*x)^2 - (1836765*(1 - 2*x))/(3 + 5*x)))/(878080*Sqrt[3 + 5*
x]*(7 + (1 - 2*x)/(3 + 5*x))^6) - (4348377*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

________________________________________________________________________________________

fricas [A]  time = 2.01, size = 146, normalized size = 0.70 \begin {gather*} -\frac {21741885 \, \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 (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12293120 \, {\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/12293120*(21741885*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*s
qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(460633945*x^5 + 1555340180*x^4 + 21081
17296*x^3 + 1428134688*x^2 + 482263920*x + 64829376)*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)

________________________________________________________________________________________

giac [B]  time = 4.08, size = 484, normalized size = 2.32 \begin {gather*} \frac {4348377}{24586240} \, \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 {161051 \, \sqrt {10} {\left (27 \, {\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} + 42840 \, {\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} + 27941760 \, {\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} - 6539187200 \, {\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} - 940423680000 \, {\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 {46467993600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {185871974400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

4348377/24586240*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 161051/87808*sqrt(10)*(27*((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 + 42840*((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 + 2794176
0*((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 - 6539187200*((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 - 940423680000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))^3 - 46467993600000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1858719744
00000*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

________________________________________________________________________________________

maple [B]  time = 0.01, size = 346, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (15849834165 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \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^{5}[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 )}{12293120 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12293120*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(15849834165*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))[phone]*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^4*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^5+93924943200*7^(1/2)*x^3*arct
an(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+46962471600*7^(1/2)*x^2*arc
tan(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+12523325760*7^(1/2)*x*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+1391480640*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+907611264*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(3*x+2)^6

________________________________________________________________________________________

maxima [A]  time = 1.57, size = 273, normalized size = 1.31 \begin {gather*} \frac {272085}{307328} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {23 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{420 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {297 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {10989 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {489753 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {6648345}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4348377}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5857731}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {645909 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1229312 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

272085/307328*(-10*x^2 - x + 3)^(3/2) - 1/42*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3
+ 2160*x^2 + 576*x + 64) + 23/420*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 3
2) + 297/1568*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 10989/21952*(-10*x^2 - x + 3)
^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 489753/614656*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 6648345/61465
6*sqrt(-10*x^2 - x + 3)*x + 4348377/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 585773
1/1229312*sqrt(-10*x^2 - x + 3) + 645909/1229312*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________