3.23.78 \(\int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\) [2278]

Optimal. Leaf size=94 \[ \frac {121}{160} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {11}{16} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {1331 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{160 \sqrt {10}} \]

[Out]

-1/6*(1-2*x)^(3/2)*(3+5*x)^(3/2)+1331/1600*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-11/16*(1-2*x)^(3/2)*(3
+5*x)^(1/2)+121/160*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {1331 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{160 \sqrt {10}}-\frac {1}{6} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {11}{16} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {121}{160} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x)^(3/2),x]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/160 - (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/
2))/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(160*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 \sqrt {1-2 x} (3+5 x)^{3/2} \, dx &=-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {11}{4} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {11}{16} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {121}{32} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {121}{160} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {11}{16} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {1331}{320} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {121}{160} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {11}{16} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {1331 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{160 \sqrt {5}}\\ &=\frac {121}{160} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {11}{16} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {1331 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{160 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.78 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-621+1185 x+6100 x^2+4000 x^3\right )-3993 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{4800 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-621 + 1185*x + 6100*x^2 + 4000*x^3) - 3993*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 +
5*x]])/(4800*Sqrt[3 + 5*x])

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Maple [A]
time = 0.16, size = 88, normalized size = 0.94

method	result	size
default	\(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{6}-\frac {11 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{16}+\frac {121 \sqrt {1-2 x}\, \sqrt {3+5 x}}{160}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3200 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(88\)
risch	\(-\frac {\left (800 x^{2}+740 x -207\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{480 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3200 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(1-2*x)^(3/2)*(3+5*x)^(3/2)-11/16*(1-2*x)^(3/2)*(3+5*x)^(1/2)+121/160*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/32
00*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 = 0.48, size = 55, normalized size = 0.59 \begin {gather*} -\frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {11}{8} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1331}{3200} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {11}{160} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-10*x^2 - x + 3)^(3/2) + 11/8*sqrt(-10*x^2 - x + 3)*x - 1331/3200*sqrt(10)*arcsin(-20/11*x - 1/11) + 11/
160*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 1.86, size = 67, normalized size = 0.71 \begin {gather*} \frac {1}{480} \, {\left (800 \, x^{2} + 740 \, x - 207\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1331}{3200} \, \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((3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/480*(800*x^2 + 740*x - 207)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/3200*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 = 4.90, size = 228, normalized size = 2.43 \begin {gather*} \begin {cases} \frac {50 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {10 x - 5}} - \frac {275 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{12 \sqrt {10 x - 5}} - \frac {121 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{48 \sqrt {10 x - 5}} + \frac {1331 i \sqrt {x + \frac {3}{5}}}{160 \sqrt {10 x - 5}} - \frac {1331 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1600} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {1331 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1600} - \frac {50 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {5 - 10 x}} + \frac {275 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{12 \sqrt {5 - 10 x}} + \frac {121 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{48 \sqrt {5 - 10 x}} - \frac {1331 \sqrt {x + \frac {3}{5}}}{160 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((50*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 275*I*(x + 3/5)**(5/2)/(12*sqrt(10*x - 5)) - 121*I*(x +
3/5)**(3/2)/(48*sqrt(10*x - 5)) + 1331*I*sqrt(x + 3/5)/(160*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110)*
sqrt(x + 3/5)/11)/1600, Abs(x + 3/5) > 11/10), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/1600 - 50*(x +
3/5)**(7/2)/(3*sqrt(5 - 10*x)) + 275*(x + 3/5)**(5/2)/(12*sqrt(5 - 10*x)) + 121*(x + 3/5)**(3/2)/(48*sqrt(5 -
10*x)) - 1331*sqrt(x + 3/5)/(160*sqrt(5 - 10*x)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (67) = 134\).
time = 0.59, size = 140, normalized size = 1.49 \begin {gather*} \frac {1}{4800} \, \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 {3}{200} \, \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 {9}{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((3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/4800*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*sq
rt(22)*sqrt(5*x + 3))) + 3/200*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) + 9/50*sqrt(5)*(11*sqrt(2)*arcsin(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 \sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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