3.2602 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{25 (5 x+3)^{3/2}}{6 \sqrt{1-2 x}}-\frac{125}{8} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{275}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-125*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8 - (25*(3 + 5*x)^(3/2))/(6*Sqrt[1 - 2*x]) + (3 + 5*x)^(5/2)/(3*(1 - 2*x)^(
3/2)) + (275*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/8

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Rubi [A]  time = 0.0208511, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {47, 50, 54, 216} \[ \frac{(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{25 (5 x+3)^{3/2}}{6 \sqrt{1-2 x}}-\frac{125}{8} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{275}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-125*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8 - (25*(3 + 5*x)^(3/2))/(6*Sqrt[1 - 2*x]) + (3 + 5*x)^(5/2)/(3*(1 - 2*x)^(
3/2)) + (275*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/8

Rule 47

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

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}}{(1-2 x)^{5/2}} \, dx &=\frac{(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{25}{6} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{25 (3+5 x)^{3/2}}{6 \sqrt{1-2 x}}+\frac{(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{125}{4} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{125}{8} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{25 (3+5 x)^{3/2}}{6 \sqrt{1-2 x}}+\frac{(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{1375}{16} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{125}{8} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{25 (3+5 x)^{3/2}}{6 \sqrt{1-2 x}}+\frac{(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{1}{8} \left (275 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{125}{8} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{25 (3+5 x)^{3/2}}{6 \sqrt{1-2 x}}+\frac{(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{275}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0077158, size = 39, normalized size = 0.41 \[ \frac{121 \sqrt{\frac{11}{2}} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(121*Sqrt[11/2]*Hypergeometric2F1[-5/2, -3/2, -1/2, (5*(1 - 2*x))/11])/(12*(1 - 2*x)^(3/2))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 3+5\,x \right ) ^{{\frac{5}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.84669, size = 174, normalized size = 1.81 \begin{align*} \frac{275}{32} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{55 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{24 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{605 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1925 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

275/32*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x +
1) - 55/24*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 605/48*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1)
+ 1925/48*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.53325, size = 282, normalized size = 2.94 \begin{align*} -\frac{825 \, \sqrt{5} \sqrt{2}{\left (4 \, x^{2} - 4 \, x + 1\right )} \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 ) + 4 \,{\left (300 \, x^{2} - 1840 \, x + 603\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(825*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) + 4*(300*x^2 - 1840*x + 603)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 16.0137, size = 729, normalized size = 7.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16500*sqrt(10)*I*(x + 3/5)**(27/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(960*(x + 3/5)
**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 8250*sqrt(10)*pi*(x + 3/5)**(27/2)*sqrt(10*
x - 5)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 18150*sqrt(10)*I*(x +
3/5)**(25/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x
+ 3/5)**(25/2)*sqrt(10*x - 5)) - 9075*sqrt(10)*pi*(x + 3/5)**(25/2)*sqrt(10*x - 5)/(960*(x + 3/5)**(27/2)*sqrt
(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 30000*I*(x + 3/5)**15/(960*(x + 3/5)**(27/2)*sqrt(10*x -
5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 220000*I*(x + 3/5)**14/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) -
1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 181500*I*(x + 3/5)**13/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(
x + 3/5)**(25/2)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (8250*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(27/2)*asi
n(sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) -
9075*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(25/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(5 - 10*x)*(x + 3/5)*
*(27/2) - 528*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) + 15000*(x + 3/5)**15/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) -
528*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) - 110000*(x + 3/5)**14/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(
5 - 10*x)*(x + 3/5)**(25/2)) + 90750*(x + 3/5)**13/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(5 - 10*x)*
(x + 3/5)**(25/2)), True))

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Giac [A]  time = 2.30505, size = 96, normalized size = 1. \begin{align*} \frac{275}{16} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \, \sqrt{5}{\left (5 \, x + 3\right )} - 110 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1815 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{120 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

275/16*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/120*(4*(3*sqrt(5)*(5*x + 3) - 110*sqrt(5))*(5*x + 3) +
1815*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2