3.24.91 \(\int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\) [2391]

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

[Out]

14641/32000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+121/960*(1-2*x)^(3/2)*(3+5*x)^(1/2)+11/240*(1-2*x)^(5
/2)*(3+5*x)^(1/2)-1/8*(1-2*x)^(7/2)*(3+5*x)^(1/2)+1331/3200*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, 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 = {52, 56, 222} \begin {gather*} \frac {14641 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}}-\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} \end {gather*}

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 52

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 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 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 (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 \text {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*}

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Mathematica [A]
time = 0.19, size = 78, normalized size = 0.67 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (13329+31275 x-22820 x^2-34400 x^3+48000 x^4\right )-43923 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{96000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(13329 + 31275*x - 22820*x^2 - 34400*x^3 + 48000*x^4) - 43923*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/
2 - 5*x]/Sqrt[3 + 5*x]])/(96000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.13, size = 104, normalized size = 0.90

method	result	size
risch	\(-\frac {\left (9600 x^{3}-12640 x^{2}+3020 x +4443\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{9600 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(103\)
default	\(\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{20}+\frac {11 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{120}+\frac {121 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{800}-\frac {1331 \sqrt {1-2 x}\, \sqrt {3+5 x}}{3200}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2
)/(3+5*x)^(1/2)

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Maxima [A]
time = 0.67, size = 70, normalized size = 0.60 \begin {gather*} -\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 {gather*}

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 = 0.49, size = 72, normalized size = 0.62 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 8.93, size = 267, normalized size = 2.30 \begin {gather*} \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}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\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 {gather*}

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, Abs(x + 3/5) > 11/10), (14641*sqrt
(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*sqr
t(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)) - 1
4641*sqrt(x + 3/5)/(3200*sqrt(5 - 10*x)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (83) = 166\).
time = 0.56, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{480000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{15000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {7}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

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 - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 18
4305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/15000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5
*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 7/2000*sqrt(5)*(2*(20*x - 23)*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/50*sqrt(5)*(11*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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

[Out]

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

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