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

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

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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 = {50, 54, 216} \begin {gather*} -\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}+\frac {1331}{64} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \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 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 \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 ) \operatorname {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 = 0.04, size = 78, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {1-2 x} \left (2 \sqrt {2 x-1} \sqrt {5 x+3} \left (800 x^2+2060 x+2763\right )+3993 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{384 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/384*(Sqrt[1 - 2*x]*(2*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(2763 + 2060*x + 800*x^2) + 3993*Sqrt[10]*ArcSinh[Sqrt[5
/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.10, size = 111, normalized size = 1.16 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {375 (1-2 x)^2}{(5 x+3)^2}+\frac {400 (1-2 x)}{5 x+3}+132\right )}{192 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}-\frac {1331}{64} \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(132 + (375*(1 - 2*x)^2)/(3 + 5*x)^2 + (400*(1 - 2*x))/(3 + 5*x)))/(192*Sqrt[3 + 5*x]*(2
+ (5*(1 - 2*x))/(3 + 5*x))^3) - (1331*Sqrt[5/2]*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/64

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fricas [A]  time = 1.43, 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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giac [A]  time = 1.01, 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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maple [A]  time = 0.00, size = 88, normalized size = 0.92 \begin {gather*} \frac {1331 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{256 \sqrt {5 x +3}\, \sqrt {-2 x +1}}-\frac {\left (5 x +3\right )^{\frac {5}{2}} \sqrt {-2 x +1}}{6}-\frac {55 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{48}-\frac {605 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(5*x+3)^(5/2)*(-2*x+1)^(1/2)-55/48*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-605/64*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+1331/
256*((-2*x+1)*(5*x+3))^(1/2)/(5*x+3)^(1/2)/(-2*x+1)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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maxima [A]  time = 1.18, 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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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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sympy [A]  time = 7.32, size = 230, normalized size = 2.40 \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}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\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, 10*Abs(x + 3/5)/11 > 1), (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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