3.2025 \(\int \frac{(2+3 x)^3 (3+5 x)^3}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{3375}{832} (1-2 x)^{13/2}+\frac{11475}{176} (1-2 x)^{11/2}-\frac{28895}{64} (1-2 x)^{9/2}+\frac{98209}{56} (1-2 x)^{7/2}-\frac{1334949}{320} (1-2 x)^{5/2}+\frac{100793}{16} (1-2 x)^{3/2}-\frac{456533}{64} \sqrt{1-2 x} \]

[Out]

(-456533*Sqrt[1 - 2*x])/64 + (100793*(1 - 2*x)^(3/2))/16 - (1334949*(1 - 2*x)^(5/2))/320 + (98209*(1 - 2*x)^(7
/2))/56 - (28895*(1 - 2*x)^(9/2))/64 + (11475*(1 - 2*x)^(11/2))/176 - (3375*(1 - 2*x)^(13/2))/832

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Rubi [A]  time = 0.0175975, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ -\frac{3375}{832} (1-2 x)^{13/2}+\frac{11475}{176} (1-2 x)^{11/2}-\frac{28895}{64} (1-2 x)^{9/2}+\frac{98209}{56} (1-2 x)^{7/2}-\frac{1334949}{320} (1-2 x)^{5/2}+\frac{100793}{16} (1-2 x)^{3/2}-\frac{456533}{64} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*Sqrt[1 - 2*x])/64 + (100793*(1 - 2*x)^(3/2))/16 - (1334949*(1 - 2*x)^(5/2))/320 + (98209*(1 - 2*x)^(7
/2))/56 - (28895*(1 - 2*x)^(9/2))/64 + (11475*(1 - 2*x)^(11/2))/176 - (3375*(1 - 2*x)^(13/2))/832

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^3 (3+5 x)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{456533}{64 \sqrt{1-2 x}}-\frac{302379}{16} \sqrt{1-2 x}+\frac{1334949}{64} (1-2 x)^{3/2}-\frac{98209}{8} (1-2 x)^{5/2}+\frac{260055}{64} (1-2 x)^{7/2}-\frac{11475}{16} (1-2 x)^{9/2}+\frac{3375}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{456533}{64} \sqrt{1-2 x}+\frac{100793}{16} (1-2 x)^{3/2}-\frac{1334949}{320} (1-2 x)^{5/2}+\frac{98209}{56} (1-2 x)^{7/2}-\frac{28895}{64} (1-2 x)^{9/2}+\frac{11475}{176} (1-2 x)^{11/2}-\frac{3375}{832} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0160205, size = 43, normalized size = 0.47 \[ -\frac{\sqrt{1-2 x} \left (1299375 x^6+6544125 x^5+14921900 x^4+20766885 x^3+20586249 x^2+17147586 x+18228666\right )}{5005} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(18228666 + 17147586*x + 20586249*x^2 + 20766885*x^3 + 14921900*x^4 + 6544125*x^5 + 1299375*x^
6))/5005

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{1299375\,{x}^{6}+6544125\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+18228666}{5005}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(1299375*x^6+6544125*x^5+14921900*x^4+20766885*x^3+20586249*x^2+17147586*x+18228666)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.26226, size = 86, normalized size = 0.93 \begin{align*} -\frac{3375}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{11475}{176} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{28895}{64} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{98209}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1334949}{320} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{100793}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{456533}{64} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/832*(-2*x + 1)^(13/2) + 11475/176*(-2*x + 1)^(11/2) - 28895/64*(-2*x + 1)^(9/2) + 98209/56*(-2*x + 1)^(7
/2) - 1334949/320*(-2*x + 1)^(5/2) + 100793/16*(-2*x + 1)^(3/2) - 456533/64*sqrt(-2*x + 1)

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Fricas [A]  time = 1.6258, size = 163, normalized size = 1.77 \begin{align*} -\frac{1}{5005} \,{\left (1299375 \, x^{6} + 6544125 \, x^{5} + 14921900 \, x^{4} + 20766885 \, x^{3} + 20586249 \, x^{2} + 17147586 \, x + 18228666\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(1299375*x^6 + 6544125*x^5 + 14921900*x^4 + 20766885*x^3 + 20586249*x^2 + 17147586*x + 18228666)*sqrt(
-2*x + 1)

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Sympy [A]  time = 55.1332, size = 82, normalized size = 0.89 \begin{align*} - \frac{3375 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{11475 \left (1 - 2 x\right )^{\frac{11}{2}}}{176} - \frac{28895 \left (1 - 2 x\right )^{\frac{9}{2}}}{64} + \frac{98209 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} - \frac{1334949 \left (1 - 2 x\right )^{\frac{5}{2}}}{320} + \frac{100793 \left (1 - 2 x\right )^{\frac{3}{2}}}{16} - \frac{456533 \sqrt{1 - 2 x}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(13/2)/832 + 11475*(1 - 2*x)**(11/2)/176 - 28895*(1 - 2*x)**(9/2)/64 + 98209*(1 - 2*x)**(7/2)
/56 - 1334949*(1 - 2*x)**(5/2)/320 + 100793*(1 - 2*x)**(3/2)/16 - 456533*sqrt(1 - 2*x)/64

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Giac [A]  time = 1.81556, size = 134, normalized size = 1.46 \begin{align*} -\frac{3375}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{11475}{176} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{28895}{64} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{98209}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1334949}{320} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{100793}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{456533}{64} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/832*(2*x - 1)^6*sqrt(-2*x + 1) - 11475/176*(2*x - 1)^5*sqrt(-2*x + 1) - 28895/64*(2*x - 1)^4*sqrt(-2*x +
1) - 98209/56*(2*x - 1)^3*sqrt(-2*x + 1) - 1334949/320*(2*x - 1)^2*sqrt(-2*x + 1) + 100793/16*(-2*x + 1)^(3/2
) - 456533/64*sqrt(-2*x + 1)