3.23.71 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=238 \[ -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{756 (3 x+2)^6}+\frac {1921 (5 x+3)^{3/2} \sqrt {1-2 x}}{1512 (3 x+2)^5}+\frac {40175505215 \sqrt {5 x+3} \sqrt {1-2 x}}{597445632 (3 x+2)}+\frac {384136145 \sqrt {5 x+3} \sqrt {1-2 x}}{42674688 (3 x+2)^2}+\frac {2199649 \sqrt {5 x+3} \sqrt {1-2 x}}{1524096 (3 x+2)^3}-\frac {443563 \sqrt {5 x+3} \sqrt {1-2 x}}{254016 (3 x+2)^4}-\frac {1891543995 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2458624 \sqrt {7}} \]

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Rubi [A]  time = 0.10, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \begin {gather*} -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{756 (3 x+2)^6}+\frac {1921 (5 x+3)^{3/2} \sqrt {1-2 x}}{1512 (3 x+2)^5}+\frac {40175505215 \sqrt {5 x+3} \sqrt {1-2 x}}{597445632 (3 x+2)}+\frac {384136145 \sqrt {5 x+3} \sqrt {1-2 x}}{42674688 (3 x+2)^2}+\frac {2199649 \sqrt {5 x+3} \sqrt {1-2 x}}{1524096 (3 x+2)^3}-\frac {443563 \sqrt {5 x+3} \sqrt {1-2 x}}{254016 (3 x+2)^4}-\frac {1891543995 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2458624 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^8,x]

[Out]

(-443563*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(254016*(2 + 3*x)^4) + (2199649*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1524096*(2
+ 3*x)^3) + (384136145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(42674688*(2 + 3*x)^2) + (40175505215*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(597445632*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(21*(2 + 3*x)^7) + (115*(1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2))/(756*(2 + 3*x)^6) + (1921*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1512*(2 + 3*x)^5) - (1891543995*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2458624*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 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}-\frac {1}{378} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {6285}{4}+1245 x\right )}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {\int \frac {\left (\frac {1131615}{8}-\frac {407325 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{5670}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {\int \frac {\frac {38989515}{16}-\frac {14249325 x}{4}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{476280}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {\int \frac {\frac {7285747875}{32}-\frac {1154815725 x}{4}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{10001880}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {\int \frac {\frac {868352079525}{64}-\frac {201671476125 x}{16}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{140026320}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}+\frac {40175505215 \sqrt {1-2 x} \sqrt {3+5 x}}{597445632 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {\int \frac {48262745032425}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{980184240}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}+\frac {40175505215 \sqrt {1-2 x} \sqrt {3+5 x}}{597445632 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {1891543995 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4917248}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}+\frac {40175505215 \sqrt {1-2 x} \sqrt {3+5 x}}{597445632 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac {1891543995 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2458624}\\ &=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}+\frac {40175505215 \sqrt {1-2 x} \sqrt {3+5 x}}{597445632 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}-\frac {1891543995 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2458624 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 221, normalized size = 0.93 \begin {gather*} \frac {1}{49} \left (\frac {47 (5 x+3)^{5/2} (1-2 x)^{7/2}}{4 (3 x+2)^6}+\frac {3 (5 x+3)^{5/2} (1-2 x)^{7/2}}{(3 x+2)^7}+\frac {783 \left (43904 (1-2 x)^{5/2} (5 x+3)^{5/2}+55 (3 x+2) \left (5488 (1-2 x)^{3/2} (5 x+3)^{5/2}+11 (3 x+2) \left (2744 \sqrt {1-2 x} (5 x+3)^{5/2}-11 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (169 x+108)+363 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )\right )\right )}{351232 (3 x+2)^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^8,x]

[Out]

((3*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7 + (47*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^6) + (783
*(43904*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2) + 55*(2 + 3*x)*(5488*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2) + 11*(2 + 3*x)*(2
744*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2) - 11*(2 + 3*x)*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(108 + 169*x) + 363*Sqrt[7]*(2
+ 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])))))/(351232*(2 + 3*x)^5))/49

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IntegrateAlgebraic [A]  time = 0.59, size = 170, normalized size = 0.71 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {11745 (1-2 x)^6}{(5 x+3)^6}+\frac {548100 (1-2 x)^5}{(5 x+3)^5}-\frac {13728379 (1-2 x)^4}{(5 x+3)^4}-\frac {117462016 (1-2 x)^3}{(5 x+3)^3}-\frac {532035189 (1-2 x)^2}{(5 x+3)^2}-\frac {1315988100 (1-2 x)}{5 x+3}-1381787505\right )}{2458624 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^7}-\frac {1891543995 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2458624 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^8,x]

[Out]

(-161051*Sqrt[1 - 2*x]*(-1381787505 + (11745*(1 - 2*x)^6)/(3 + 5*x)^6 + (548100*(1 - 2*x)^5)/(3 + 5*x)^5 - (13
728379*(1 - 2*x)^4)/(3 + 5*x)^4 - (117462016*(1 - 2*x)^3)/(3 + 5*x)^3 - (532035189*(1 - 2*x)^2)/(3 + 5*x)^2 -
(1315988100*(1 - 2*x))/(3 + 5*x)))/(2458624*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^7) - (1891543995*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2458624*Sqrt[7])

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fricas [A]  time = 0.76, size = 161, normalized size = 0.68 \begin {gather*} -\frac {1891543995 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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 (120526515645 \, x^{6} + 487483968610 \, x^{5} + 821723878536 \, x^{4} + 738910550592 \, x^{3} + 373848853744 \, x^{2} + 100906793184 \, x + 11351210112\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{34420736 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/34420736*(1891543995*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x
+ 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(120526515645*x^6 +
487483968610*x^5 + 821723878536*x^4 + 738910550592*x^3 + 373848853744*x^2 + 100906793184*x + 11351210112)*sqr
t(5*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128
)

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giac [B]  time = 5.49, size = 542, normalized size = 2.28 \begin {gather*} \frac {378308799}{68841472} \, \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 {805255 \, \sqrt {10} {\left (2349 \, {\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 )}^{13} + 4384800 \, {\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} - 4393081280 \, {\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} - 1503513804800 \, {\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} - 272402016768000 \, {\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} - 26951436288000000 \, {\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 {1131960324096000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {4527841296384000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1229312 \, {\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 )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="giac")

[Out]

378308799/68841472*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)))) - 805255/1229312*sqrt(10)*(2349*((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)))^13 + 4384800*(
(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
- 4393081280*((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 - 1503513804800*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^7 - 272402016768000*((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 - 26951436288000000*((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 - 1131960324096000000*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) + 4527841296384000000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqr
t(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
)^7

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maple [B]  time = 0.01, size = 394, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4136806717065 \sqrt {7}\, x^{7} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{6} \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^{6}[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 )}{34420736 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^(3/2)/(3*x+2)^8,x)

[Out]

1/34420736*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(4136806717065*7^(1/2)*x^7*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))[phone]*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]*(-10*x^2-x+3)^(1/2)*x^6+42900217806600*7^(
1/2)*x^4*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^5+286001452044
00*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+11440
058081760*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+2542235129280*7^(1/2)*x*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^2+242117631360*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+158916941568*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^7

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maxima [A]  time = 1.33, size = 324, normalized size = 1.36 \begin {gather*} \frac {118356975}{4302592} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{7 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {305 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{588 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {2161 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1176 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {129195 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {4780215 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{307328 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {213042555 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8605184 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {2892030075}{8605184} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1891543995}{34420736} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2548112985}{17210368} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {280970415 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{17210368 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="maxima")

[Out]

118356975/4302592*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22
680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 305/588*(-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) + 2161/1176*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*
x^2 + 240*x + 32) + 129195/21952*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 4780215/30
7328*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 213042555/8605184*(-10*x^2 - x + 3)^(5/2)/(9*x^2 +
12*x + 4) + 2892030075/8605184*sqrt(-10*x^2 - x + 3)*x + 1891543995/34420736*sqrt(7)*arcsin(37/11*x/abs(3*x +
2) + 20/11/abs(3*x + 2)) - 2548112985/17210368*sqrt(-10*x^2 - x + 3) + 280970415/17210368*(-10*x^2 - x + 3)^(
3/2)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^8,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^8, x)

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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)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**8,x)

[Out]

Timed out

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