3.2087 \(\int \frac {(2+3 x) (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {15}{8} (1-2 x)^{5/2}-\frac {505}{24} (1-2 x)^{3/2}+\frac {1133}{8} \sqrt {1-2 x}+\frac {847}{8 \sqrt {1-2 x}} \]

[Out]

-505/24*(1-2*x)^(3/2)+15/8*(1-2*x)^(5/2)+847/8/(1-2*x)^(1/2)+1133/8*(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ \frac {15}{8} (1-2 x)^{5/2}-\frac {505}{24} (1-2 x)^{3/2}+\frac {1133}{8} \sqrt {1-2 x}+\frac {847}{8 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

847/(8*Sqrt[1 - 2*x]) + (1133*Sqrt[1 - 2*x])/8 - (505*(1 - 2*x)^(3/2))/24 + (15*(1 - 2*x)^(5/2))/8

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x) (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {847}{8 (1-2 x)^{3/2}}-\frac {1133}{8 \sqrt {1-2 x}}+\frac {505}{8} \sqrt {1-2 x}-\frac {75}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {847}{8 \sqrt {1-2 x}}+\frac {1133}{8} \sqrt {1-2 x}-\frac {505}{24} (1-2 x)^{3/2}+\frac {15}{8} (1-2 x)^{5/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \[ \frac {-45 x^3-185 x^2-631 x+685}{3 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(685 - 631*x - 185*x^2 - 45*x^3)/(3*Sqrt[1 - 2*x])

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 31, normalized size = 0.58 \[ \frac {{\left (45 \, x^{3} + 185 \, x^{2} + 631 \, x - 685\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(45*x^3 + 185*x^2 + 631*x - 685)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

giac [A]  time = 1.20, size = 44, normalized size = 0.83 \[ \frac {15}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {505}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1133}{8} \, \sqrt {-2 \, x + 1} + \frac {847}{8 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/8*(2*x - 1)^2*sqrt(-2*x + 1) - 505/24*(-2*x + 1)^(3/2) + 1133/8*sqrt(-2*x + 1) + 847/8/sqrt(-2*x + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.47 \[ -\frac {45 x^{3}+185 x^{2}+631 x -685}{3 \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(45*x^3+185*x^2+631*x-685)/(-2*x+1)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 37, normalized size = 0.70 \[ \frac {15}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {505}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1133}{8} \, \sqrt {-2 \, x + 1} + \frac {847}{8 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/8*(-2*x + 1)^(5/2) - 505/24*(-2*x + 1)^(3/2) + 1133/8*sqrt(-2*x + 1) + 847/8/sqrt(-2*x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \[ \frac {847}{8\,\sqrt {1-2\,x}}+\frac {1133\,\sqrt {1-2\,x}}{8}-\frac {505\,{\left (1-2\,x\right )}^{3/2}}{24}+\frac {15\,{\left (1-2\,x\right )}^{5/2}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

847/(8*(1 - 2*x)^(1/2)) + (1133*(1 - 2*x)^(1/2))/8 - (505*(1 - 2*x)^(3/2))/24 + (15*(1 - 2*x)^(5/2))/8

________________________________________________________________________________________

sympy [A]  time = 17.66, size = 46, normalized size = 0.87 \[ \frac {15 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {505 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} + \frac {1133 \sqrt {1 - 2 x}}{8} + \frac {847}{8 \sqrt {1 - 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*(1 - 2*x)**(5/2)/8 - 505*(1 - 2*x)**(3/2)/24 + 1133*sqrt(1 - 2*x)/8 + 847/(8*sqrt(1 - 2*x))

________________________________________________________________________________________