3.2027 \(\int \frac{(2+3 x) (3+5 x)^3}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{125}{48} (1-2 x)^{9/2}+\frac{1675}{56} (1-2 x)^{7/2}-\frac{561}{4} (1-2 x)^{5/2}+\frac{2783}{8} (1-2 x)^{3/2}-\frac{9317}{16} \sqrt{1-2 x} \]

[Out]

(-9317*Sqrt[1 - 2*x])/16 + (2783*(1 - 2*x)^(3/2))/8 - (561*(1 - 2*x)^(5/2))/4 + (1675*(1 - 2*x)^(7/2))/56 - (1
25*(1 - 2*x)^(9/2))/48

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Rubi [A]  time = 0.0116112, antiderivative size = 66, 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} \[ -\frac{125}{48} (1-2 x)^{9/2}+\frac{1675}{56} (1-2 x)^{7/2}-\frac{561}{4} (1-2 x)^{5/2}+\frac{2783}{8} (1-2 x)^{3/2}-\frac{9317}{16} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*Sqrt[1 - 2*x])/16 + (2783*(1 - 2*x)^(3/2))/8 - (561*(1 - 2*x)^(5/2))/4 + (1675*(1 - 2*x)^(7/2))/56 - (1
25*(1 - 2*x)^(9/2))/48

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)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{9317}{16 \sqrt{1-2 x}}-\frac{8349}{8} \sqrt{1-2 x}+\frac{2805}{4} (1-2 x)^{3/2}-\frac{1675}{8} (1-2 x)^{5/2}+\frac{375}{16} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{9317}{16} \sqrt{1-2 x}+\frac{2783}{8} (1-2 x)^{3/2}-\frac{561}{4} (1-2 x)^{5/2}+\frac{1675}{56} (1-2 x)^{7/2}-\frac{125}{48} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0115998, size = 33, normalized size = 0.5 \[ -\frac{1}{21} \sqrt{1-2 x} \left (875 x^4+3275 x^3+5556 x^2+6161 x+7295\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(7295 + 6161*x + 5556*x^2 + 3275*x^3 + 875*x^4))/21

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{875\,{x}^{4}+3275\,{x}^{3}+5556\,{x}^{2}+6161\,x+7295}{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)^3/(1-2*x)^(1/2),x)

[Out]

-1/21*(875*x^4+3275*x^3+5556*x^2+6161*x+7295)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.89769, size = 62, normalized size = 0.94 \begin{align*} -\frac{125}{48} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1675}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{561}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2783}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{9317}{16} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/48*(-2*x + 1)^(9/2) + 1675/56*(-2*x + 1)^(7/2) - 561/4*(-2*x + 1)^(5/2) + 2783/8*(-2*x + 1)^(3/2) - 9317/
16*sqrt(-2*x + 1)

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Fricas [A]  time = 1.57697, size = 95, normalized size = 1.44 \begin{align*} -\frac{1}{21} \,{\left (875 \, x^{4} + 3275 \, x^{3} + 5556 \, x^{2} + 6161 \, x + 7295\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)^3/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/21*(875*x^4 + 3275*x^3 + 5556*x^2 + 6161*x + 7295)*sqrt(-2*x + 1)

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Sympy [A]  time = 30.0102, size = 58, normalized size = 0.88 \begin{align*} - \frac{125 \left (1 - 2 x\right )^{\frac{9}{2}}}{48} + \frac{1675 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} - \frac{561 \left (1 - 2 x\right )^{\frac{5}{2}}}{4} + \frac{2783 \left (1 - 2 x\right )^{\frac{3}{2}}}{8} - \frac{9317 \sqrt{1 - 2 x}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(9/2)/48 + 1675*(1 - 2*x)**(7/2)/56 - 561*(1 - 2*x)**(5/2)/4 + 2783*(1 - 2*x)**(3/2)/8 - 9317*
sqrt(1 - 2*x)/16

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Giac [A]  time = 1.9422, size = 90, normalized size = 1.36 \begin{align*} -\frac{125}{48} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1675}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{561}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2783}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{9317}{16} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/48*(2*x - 1)^4*sqrt(-2*x + 1) - 1675/56*(2*x - 1)^3*sqrt(-2*x + 1) - 561/4*(2*x - 1)^2*sqrt(-2*x + 1) + 2
783/8*(-2*x + 1)^(3/2) - 9317/16*sqrt(-2*x + 1)