3.2391 \(\int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{8} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{11}{240} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{960} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{1331 \sqrt{5 x+3} \sqrt{1-2 x}}{3200}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/960 + (11*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/240 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/8 + (14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi [A]  time = 0.0307865, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{8} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{11}{240} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{960} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{1331 \sqrt{5 x+3} \sqrt{1-2 x}}{3200}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*Sqrt[3 + 5*x],x]

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/960 + (11*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/240 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/8 + (14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

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 216

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 (1-2 x)^{5/2} \sqrt{3+5 x} \, dx &=-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{11}{16} \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{11}{240} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{121}{96} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{121}{960} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{11}{240} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{1331}{640} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{1331 \sqrt{1-2 x} \sqrt{3+5 x}}{3200}+\frac{121}{960} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{11}{240} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{14641 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{6400}\\ &=\frac{1331 \sqrt{1-2 x} \sqrt{3+5 x}}{3200}+\frac{121}{960} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{11}{240} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{14641 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{3200 \sqrt{5}}\\ &=\frac{1331 \sqrt{1-2 x} \sqrt{3+5 x}}{3200}+\frac{121}{960} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{11}{240} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{3200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0551464, size = 74, normalized size = 0.64 \[ -\frac{10 \sqrt{5 x+3} \left (19200 x^4-34880 x^3+18680 x^2+5866 x-4443\right )+43923 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{96000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(-4443 + 5866*x + 18680*x^2 - 34880*x^3 + 19200*x^4) + 43923*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(96000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{20} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{11}{120} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{121}{800} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{3200}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{14641\,\sqrt{10}}{64000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(1-2*x)^(5/2)*(3+5*x)^(3/2)+11/120*(1-2*x)^(3/2)*(3+5*x)^(3/2)+121/800*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1331/3
200*(1-2*x)^(1/2)*(3+5*x)^(1/2)+14641/64000*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsi
n(20/11*x+1/11)

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Maxima [A]  time = 2.18491, size = 95, normalized size = 0.82 \begin{align*} -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{17}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{121}{160} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{14641}{64000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{121}{3200} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(-10*x^2 - x + 3)^(3/2)*x + 17/120*(-10*x^2 - x + 3)^(3/2) + 121/160*sqrt(-10*x^2 - x + 3)*x - 14641/640
00*sqrt(10)*arcsin(-20/11*x - 1/11) + 121/3200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.45509, size = 240, normalized size = 2.07 \begin{align*} \frac{1}{9600} \,{\left (9600 \, x^{3} - 12640 \, x^{2} + 3020 \, x + 4443\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{14641}{64000} \, \sqrt{10} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9600*(9600*x^3 - 12640*x^2 + 3020*x + 4443)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 14641/64000*sqrt(10)*arctan(1/20*
sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]  time = 17.1593, size = 269, normalized size = 2.32 \begin{align*} \begin{cases} \frac{10 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{10 x - 5}} - \frac{253 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{6 \sqrt{10 x - 5}} + \frac{15367 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{240 \sqrt{10 x - 5}} - \frac{177023 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4800 \sqrt{10 x - 5}} + \frac{14641 i \sqrt{x + \frac{3}{5}}}{3200 \sqrt{10 x - 5}} - \frac{14641 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{32000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{32000} - \frac{10 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{5 - 10 x}} + \frac{253 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{6 \sqrt{5 - 10 x}} - \frac{15367 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{240 \sqrt{5 - 10 x}} + \frac{177023 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4800 \sqrt{5 - 10 x}} - \frac{14641 \sqrt{x + \frac{3}{5}}}{3200 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((10*I*(x + 3/5)**(9/2)/sqrt(10*x - 5) - 253*I*(x + 3/5)**(7/2)/(6*sqrt(10*x - 5)) + 15367*I*(x + 3/5
)**(5/2)/(240*sqrt(10*x - 5)) - 177023*I*(x + 3/5)**(3/2)/(4800*sqrt(10*x - 5)) + 14641*I*sqrt(x + 3/5)/(3200*
sqrt(10*x - 5)) - 14641*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/32000, 10*Abs(x + 3/5)/11 > 1), (14641*sq
rt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/32000 - 10*(x + 3/5)**(9/2)/sqrt(5 - 10*x) + 253*(x + 3/5)**(7/2)/(6*s
qrt(5 - 10*x)) - 15367*(x + 3/5)**(5/2)/(240*sqrt(5 - 10*x)) + 177023*(x + 3/5)**(3/2)/(4800*sqrt(5 - 10*x)) -
14641*sqrt(x + 3/5)/(3200*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.04839, size = 220, normalized size = 1.9 \begin{align*} \frac{1}{480000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4537
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/6000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3
)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/400*sqrt(5)*(2*(20*x + 1)*sqrt(5*x +
3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))