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]

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

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Rubi [A]  time = 0.0099816, antiderivative size = 53, normalized size of antiderivative = 1., 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 = 0.0101174, 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])

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{45\,{x}^{3}+185\,{x}^{2}+631\,x-685}{3}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.23728, size = 50, normalized size = 0.94 \begin{align*} \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}} \end{align*}

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)

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Fricas [A]  time = 1.59331, size = 85, normalized size = 1.6 \begin{align*} \frac{{\left (45 \, x^{3} + 185 \, x^{2} + 631 \, x - 685\right )} \sqrt{-2 \, x + 1}}{3 \,{\left (2 \, x - 1\right )}} \end{align*}

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)

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Sympy [A]  time = 11.6723, size = 46, normalized size = 0.87 \begin{align*} \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}} \end{align*}

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))

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Giac [A]  time = 2.27465, size = 59, normalized size = 1.11 \begin{align*} \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}} \end{align*}

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)