3.2016 \(\int \frac{(2+3 x) (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]

[Out]

(-847*Sqrt[1 - 2*x])/8 + (1133*(1 - 2*x)^(3/2))/24 - (101*(1 - 2*x)^(5/2))/8 + (75*(1 - 2*x)^(7/2))/56

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Rubi [A]  time = 0.0106217, 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} \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-847*Sqrt[1 - 2*x])/8 + (1133*(1 - 2*x)^(3/2))/24 - (101*(1 - 2*x)^(5/2))/8 + (75*(1 - 2*x)^(7/2))/56

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}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{847}{8 \sqrt{1-2 x}}-\frac{1133}{8} \sqrt{1-2 x}+\frac{505}{8} (1-2 x)^{3/2}-\frac{75}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{847}{8} \sqrt{1-2 x}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{75}{56} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0101598, size = 28, normalized size = 0.53 \[ -\frac{1}{21} \sqrt{1-2 x} \left (225 x^3+723 x^2+1091 x+1469\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(1469 + 1091*x + 723*x^2 + 225*x^3))/21

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Maple [A]  time = 0.002, size = 25, normalized size = 0.5 \begin{align*} -{\frac{225\,{x}^{3}+723\,{x}^{2}+1091\,x+1469}{21}\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)^(1/2),x)

[Out]

-1/21*(225*x^3+723*x^2+1091*x+1469)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.01415, size = 50, normalized size = 0.94 \begin{align*} \frac{75}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{101}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="maxima")

[Out]

75/56*(-2*x + 1)^(7/2) - 101/8*(-2*x + 1)^(5/2) + 1133/24*(-2*x + 1)^(3/2) - 847/8*sqrt(-2*x + 1)

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Fricas [A]  time = 1.52172, size = 78, normalized size = 1.47 \begin{align*} -\frac{1}{21} \,{\left (225 \, x^{3} + 723 \, x^{2} + 1091 \, x + 1469\right )} \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)^(1/2),x, algorithm="fricas")

[Out]

-1/21*(225*x^3 + 723*x^2 + 1091*x + 1469)*sqrt(-2*x + 1)

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Sympy [A]  time = 20.4082, size = 46, normalized size = 0.87 \begin{align*} \frac{75 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} - \frac{101 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} + \frac{1133 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{847 \sqrt{1 - 2 x}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*(1 - 2*x)**(7/2)/56 - 101*(1 - 2*x)**(5/2)/8 + 1133*(1 - 2*x)**(3/2)/24 - 847*sqrt(1 - 2*x)/8

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Giac [A]  time = 1.60995, size = 69, normalized size = 1.3 \begin{align*} -\frac{75}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{101}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="giac")

[Out]

-75/56*(2*x - 1)^3*sqrt(-2*x + 1) - 101/8*(2*x - 1)^2*sqrt(-2*x + 1) + 1133/24*(-2*x + 1)^(3/2) - 847/8*sqrt(-
2*x + 1)