3.2593 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{5 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{5}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-5*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) + (5*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/2

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Rubi [A]  time = 0.0142685, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {47, 54, 216} \[ \frac{(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{5 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{5}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) + (5*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/2

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

Mathematica [C]  time = 0.0076622, size = 39, normalized size = 0.53 \[ \frac{11 \sqrt{\frac{11}{2}} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{6 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*Sqrt[11/2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (5*(1 - 2*x))/11])/(6*(1 - 2*x)^(3/2))

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 3+5\,x \right ) ^{{\frac{3}{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)^(3/2)/(1-2*x)^(5/2),x)

[Out]

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

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Maxima [A]  time = 3.69398, size = 126, normalized size = 1.7 \begin{align*} \frac{5}{8} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{35 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/6*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 11/12*sq
rt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 35/12*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.51985, size = 262, normalized size = 3.54 \begin{align*} -\frac{15 \, \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 (40 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{24 \,{\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)^(3/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/24*(15*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*(40*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 6.31686, size = 636, normalized size = 8.59 \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)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Piecewise((-300*sqrt(10)*I*(x + 3/5)**(15/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(240*(x + 3/5)**
(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 150*sqrt(10)*pi*(x + 3/5)**(15/2)*sqrt(10*x -
5)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 330*sqrt(10)*I*(x + 3/5)**(
13/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**
(13/2)*sqrt(10*x - 5)) - 165*sqrt(10)*pi*(x + 3/5)**(13/2)*sqrt(10*x - 5)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5
) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 4000*I*(x + 3/5)**8/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x
+ 3/5)**(13/2)*sqrt(10*x - 5)) - 3300*I*(x + 3/5)**7/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(
13/2)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (150*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(15/2)*asin(sqrt(110)*
sqrt(x + 3/5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x + 3/5)**(13/2)) - 165*sqrt(10)
*sqrt(5 - 10*x)*(x + 3/5)**(13/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132
*sqrt(5 - 10*x)*(x + 3/5)**(13/2)) - 2000*(x + 3/5)**8/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10
*x)*(x + 3/5)**(13/2)) + 1650*(x + 3/5)**7/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x + 3/5
)**(13/2)), True))

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Giac [A]  time = 2.49736, size = 78, normalized size = 1.05 \begin{align*} \frac{5}{4} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{30 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/30*(8*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(
-10*x + 5)/(2*x - 1)^2