3.23.94 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx\) [2294]

Optimal. Leaf size=149 \[ -\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}-\frac {50}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{31752 \sqrt {7}} \]

[Out]

-250433/222264*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-50/81*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2
))*10^(1/2)-59/252*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-1/9*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3-6401/10584*
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.04, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 154, 163, 56, 222, 95, 210} \begin {gather*} -\frac {50}{81} \sqrt {10} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {250433 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{31752 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^2}-\frac {6401 \sqrt {1-2 x} \sqrt {5 x+3}}{10584 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]

[Out]

(-6401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10584*(2 + 3*x)) - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(252*(2 + 3*x)^2) -
(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) - (50*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (250433
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(31752*Sqrt[7])

Rule 56

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 99

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 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 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{378} \int \frac {\left (\frac {801}{4}-2100 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {\int \frac {-\frac {141567}{8}-73500 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{7938}\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}-\frac {250}{81} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {250433 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{63504}\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {250433 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{31752}-\frac {1}{81} \left (100 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}-\frac {50}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{31752 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 103, normalized size = 0.69 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (51056+159174 x+124179 x^2\right )}{(2+3 x)^3}+137200 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-250433 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{222264} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]

[Out]

((-21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(51056 + 159174*x + 124179*x^2))/(2 + 3*x)^3 + 137200*Sqrt[10]*ArcTan[Sqrt[5
/2 - 5*x]/Sqrt[3 + 5*x]] - 250433*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/222264

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(113)=226\).
time = 0.13, size = 253, normalized size = 1.70

method	result	size
risch	\(\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (124179 x^{2}+159174 x +51056\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10584 \left (2+3 x \right )^{3} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {25 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{81}-\frac {250433 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{444528}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(138\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (6761691 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-3704400 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-7408800 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+9015588 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -4939200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -5215518 x^{2} \sqrt {-10 x^{2}-x +3}+2003464 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1097600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-6685308 x \sqrt {-10 x^{2}-x +3}-2144352 \sqrt {-10 x^{2}-x +3}\right )}{444528 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/444528*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(6761691*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-3
704400*10^(1/2)*arcsin(20/11*x+1/11)*x^3+13523382*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
^2-7408800*10^(1/2)*arcsin(20/11*x+1/11)*x^2+9015588*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x-4939200*10^(1/2)*arcsin(20/11*x+1/11)*x-5215518*x^2*(-10*x^2-x+3)^(1/2)+2003464*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1097600*10^(1/2)*arcsin(20/11*x+1/11)-6685308*x*(-10*x^2-x+3)^(1/2)-2144352*(
-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]
time = 0.50, size = 132, normalized size = 0.89 \begin {gather*} -\frac {25}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {250433}{444528} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {515}{2646} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{63 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {103 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {5989 \, \sqrt {-10 \, x^{2} - x + 3}}{10584 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

-25/81*sqrt(10)*arcsin(20/11*x + 1/11) + 250433/444528*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2
)) - 515/2646*sqrt(-10*x^2 - x + 3) + 1/63*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) - 103/588*(-10
*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 5989/10584*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.75, size = 156, normalized size = 1.05 \begin {gather*} -\frac {250433 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 ) - 137200 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (124179 \, x^{2} + 159174 \, x + 51056\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{444528 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/444528*(250433*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 137200*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*
x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(124179*x^2 + 159174*x + 51056)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27
*x^3 + 54*x^2 + 36*x + 8)

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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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (113) = 226\).
time = 1.89, size = 377, normalized size = 2.53 \begin {gather*} \frac {250433}{4445280} \, \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 {25}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (6401 \, {\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} + 4674880 \, {\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 {1034801600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {4139206400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{5292 \, {\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

250433/4445280*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)))) - 25/81*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/5292*sqr
t(10)*(6401*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))^5 + 4674880*((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 + 1034801600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4139206400*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)^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^4,x)

[Out]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^4, x)

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