3.25.81 \(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [2481]

Optimal. Leaf size=96 \[ -\frac {605}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1331}{64} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

[Out]

1331/128*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-55/48*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1/6*(3+5*x)^(5/2)*(1-2
*x)^(1/2)-605/64*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 96, 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}{64} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {55}{48} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {605}{64} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/Sqrt[1 - 2*x],x]

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (55*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/48 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/6 + (1331*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/64

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 \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {55}{12} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {605}{32} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {605}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {6655}{128} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {605}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{64} \left (1331 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {605}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1331}{64} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 73, normalized size = 0.76 \begin {gather*} \frac {-2 \sqrt {1-2 x} \left (8289+19995 x+12700 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 )}{384 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(8289 + 19995*x + 12700*x^2 + 4000*x^3) - 3993*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3
+ 5*x]])/(384*Sqrt[3 + 5*x])

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Maple [A]
time = 0.08, size = 88, normalized size = 0.92

method	result	size
default	\(-\frac {\left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{6}-\frac {55 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{48}-\frac {605 \sqrt {1-2 x}\, \sqrt {3+5 x}}{64}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(88\)
risch	\(\frac {\left (800 x^{2}+2060 x +2763\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{192 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3+5*x)^(5/2)*(1-2*x)^(1/2)-55/48*(3+5*x)^(3/2)*(1-2*x)^(1/2)-605/64*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/256
*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 = 0.55, size = 58, normalized size = 0.60 \begin {gather*} -\frac {25}{6} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {515}{48} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1331}{256} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {921}{64} \, \sqrt {-10 \, x^{2} - x + 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),x, algorithm="maxima")

[Out]

-25/6*sqrt(-10*x^2 - x + 3)*x^2 - 515/48*sqrt(-10*x^2 - x + 3)*x - 1331/256*sqrt(10)*arcsin(-20/11*x - 1/11) -
921/64*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.39, size = 73, normalized size = 0.76 \begin {gather*} -\frac {1}{192} \, {\left (800 \, x^{2} + 2060 \, x + 2763\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1331}{256} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\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)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/192*(800*x^2 + 2060*x + 2763)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/256*sqrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*s
qrt(2)*(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 = 9.02, size = 228, normalized size = 2.38 \begin {gather*} \begin {cases} - \frac {125 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}}}{24 \sqrt {10 x - 5}} - \frac {3025 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{96 \sqrt {10 x - 5}} + \frac {6655 i \sqrt {x + \frac {3}{5}}}{64 \sqrt {10 x - 5}} - \frac {1331 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{128} & \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 )}}{128} + \frac {125 \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}}}{24 \sqrt {5 - 10 x}} + \frac {3025 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{96 \sqrt {5 - 10 x}} - \frac {6655 \sqrt {x + \frac {3}{5}}}{64 \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)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-125*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 275*I*(x + 3/5)**(5/2)/(24*sqrt(10*x - 5)) - 3025*I*(x
+ 3/5)**(3/2)/(96*sqrt(10*x - 5)) + 6655*I*sqrt(x + 3/5)/(64*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110
)*sqrt(x + 3/5)/11)/128, Abs(x + 3/5) > 11/10), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/128 + 125*(x +
3/5)**(7/2)/(3*sqrt(5 - 10*x)) + 275*(x + 3/5)**(5/2)/(24*sqrt(5 - 10*x)) + 3025*(x + 3/5)**(3/2)/(96*sqrt(5
- 10*x)) - 6655*sqrt(x + 3/5)/(64*sqrt(5 - 10*x)), True))

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Giac [A]
time = 1.04, size = 54, normalized size = 0.56 \begin {gather*} -\frac {1}{1920} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x + 79\right )} {\left (5 \, x + 3\right )} + 1815\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 19965 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\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),x, algorithm="giac")

[Out]

-1/1920*sqrt(5)*(2*(4*(40*x + 79)*(5*x + 3) + 1815)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 19965*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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