3.25.34 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx\) [2434]

Optimal. Leaf size=180 \[ \frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \]

[Out]

-104040277/43904*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+7/15*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3
*x)^5+2023/360*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+67187/2160*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+2347559/
12096*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+245529161/169344*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.04, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 154, 156, 12, 95, 210} \begin {gather*} -\frac {104040277 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}}+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {245529161 \sqrt {5 x+3} \sqrt {1-2 x}}{169344 (3 x+2)}+\frac {2347559 \sqrt {5 x+3} \sqrt {1-2 x}}{12096 (3 x+2)^2}+\frac {67187 \sqrt {5 x+3} \sqrt {1-2 x}}{2160 (3 x+2)^3}+\frac {2023 \sqrt {5 x+3} \sqrt {1-2 x}}{360 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (2023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(360*(2 + 3*x)^4) + (6
7187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2160*(2 + 3*x)^3) + (2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^
2) + (245529161*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(169344*(2 + 3*x)) - (104040277*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(6272*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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 154

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] && ILtQ[m, -1] && GtQ[n, 0]

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 {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (\frac {421}{2}-190 x\right ) \sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}-\frac {1}{180} \int \frac {-\frac {79903}{4}+28825 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}-\frac {\int \frac {-\frac {14846615}{8}+2351545 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{3780}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}-\frac {\int \frac {-\frac {1768979345}{16}+\frac {410822825 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{52920}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {\int -\frac {98318061765}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{370440}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}+\frac {104040277 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}+\frac {104040277 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {104040277 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2341358496+13788819736 x+30475811404 x^2+29956486710 x^3+11048812245 x^4\right )}{(2+3 x)^5}-1560604155 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{658560} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2341358496 + 13788819736*x + 30475811404*x^2 + 29956486710*x^3 + 11048812245*
x^4))/(2 + 3*x)^5 - 1560604155*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/658560

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(141)=282\).
time = 0.14, size = 298, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (11048812245 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{94080 \left (2+3 x \right )^{5} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {104040277 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{87808 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(134\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (379226809665 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{1317120 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1317120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(379226809665*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^5+1264089365550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1685452487400*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+154683371430*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+419390813940*x^3*(-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+426661359656*x^2*(-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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+
3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]
time = 0.57, size = 184, normalized size = 1.02 \begin {gather*} \frac {104040277}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{45 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {637 \, \sqrt {-10 \, x^{2} - x + 3}}{120 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {67187 \, \sqrt {-10 \, x^{2} - x + 3}}{2160 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {2347559 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {245529161 \, \sqrt {-10 \, x^{2} - x + 3}}{169344 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

104040277/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/45*sqrt(-10*x^2 - x + 3)/(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 637/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) + 67187/2160*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 2347559/12096*sqrt(-10*x^2 - x +
3)/(9*x^2 + 12*x + 4) + 245529161/169344*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.77, size = 131, normalized size = 0.73 \begin {gather*} -\frac {1560604155 \, \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 (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1317120*(1560604155*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*(11048812245*x^4 + 29956486710*x^3 + 30475811404*x^
2 + 13788819736*x + 2341358496)*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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Sympy [F(-1)] Timed out
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)/(2+3*x)**6/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
time = 0.80, size = 426, normalized size = 2.37 \begin {gather*} \frac {104040277}{878080} \, \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 (706299 \, {\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} + 493892560 \, {\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} + 156884295680 \, {\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} + 24022907776000 \, {\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 {1441374466560000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5765497866240000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9408 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

104040277/878080*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)))) + 1331/9408*sqrt(10)*(706299*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 493892560*((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)))^7 + 156
884295680*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^5 + 24022907776000*((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 + 1441374466560000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 57654978662
40000*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)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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