3.24.58 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\) [2358]

Optimal. Leaf size=180 \[ -\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]

[Out]

1/5*(1-2*x)^(3/2)*(3+5*x)^(7/2)/(2+3*x)^5-483153/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2
)-1331/3136*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-121/560*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+33/40*(3+5*x)^
(7/2)*(1-2*x)^(1/2)/(2+3*x)^4-43923/43904*(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 = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210} \begin {gather*} -\frac {483153 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}}+\frac {33 \sqrt {1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac {121 \sqrt {1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac {43923 \sqrt {1-2 x} \sqrt {5 x+3}}{43904 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(3136*(2 + 3*x)^
2) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(560*(2 + 3*x)^3) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(5*(2 + 3*x)^5)
+ (33*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(43904*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 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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33}{10} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {363}{80} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {1331}{224} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {43923 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{6272}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {483153 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {483153 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac {483153 \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 = 0.30, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3507552+21437032 x+47906548 x^2+46076650 x^3+15899035 x^4\right )}{(2+3 x)^5}-2415765 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1536640} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3507552 + 21437032*x + 47906548*x^2 + 46076650*x^3 + 15899035*x^4))/(2 + 3*x)
^5 - 2415765*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1536640

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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 = 0.15, size = 298, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (15899035 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +3507552\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{219520 \left (2+3 x \right )^{5} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {483153 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(134\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (587030895 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(587030895*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
5+1956769650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2609026200*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+222586490*x^4*(-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^2+645073100*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+670691672*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 = 0.50, size = 227, normalized size = 1.26 \begin {gather*} \frac {90695}{230496} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{392 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1221 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5488 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {54417 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {738705}{153664} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {483153}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {650859}{307328} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {215303 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{921984 \, {\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)^6,x, algorithm="maxima")

[Out]

90695/230496*(-10*x^2 - x + 3)^(3/2) - 1/35*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 +
240*x + 32) + 33/392*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1221/5488*(-10*x^2 - x
+ 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 54417/153664*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 738705/15
3664*sqrt(-10*x^2 - x + 3)*x + 483153/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 65085
9/307328*sqrt(-10*x^2 - x + 3) + 215303/921984*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]
time = 1.08, size = 131, normalized size = 0.73 \begin {gather*} -\frac {2415765 \, \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 (15899035 \, x^{4} + 46076650 \, x^{3} + 47906548 \, x^{2} + 21437032 \, x + 3507552\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 )}} \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)^6,x, algorithm="fricas")

[Out]

-1/3073280*(2415765*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*(15899035*x^4 + 46076650*x^3 + 47906548*x^2 + 21437032
*x + 3507552)*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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**6,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 = 1.69, size = 426, normalized size = 2.37 \begin {gather*} \frac {483153}{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 {161051 \, \sqrt {10} {\left (3 \, {\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} + 3920 \, {\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} + 2007040 \, {\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} - 307328000 \, {\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 {18439680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {73758720000 \, \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}} \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)^6,x, algorithm="giac")

[Out]

483153/6146560*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)))) - 161051/21952*sqrt(10)*(3*((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 + 3920*((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 + 2007040*((sq
rt(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 - 30
7328000*((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 - 18439680000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 73758720000*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)*s
qrt(-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 )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,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)^6,x)

[Out]

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

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