3.2159 \(\int \frac{(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{375}{64} (1-2 x)^{9/2}+\frac{11475}{112} (1-2 x)^{7/2}-\frac{52011}{64} (1-2 x)^{5/2}+\frac{98209}{24} (1-2 x)^{3/2}-\frac{1334949}{64} \sqrt{1-2 x}-\frac{302379}{16 \sqrt{1-2 x}}+\frac{456533}{192 (1-2 x)^{3/2}} \]

[Out]

456533/(192*(1 - 2*x)^(3/2)) - 302379/(16*Sqrt[1 - 2*x]) - (1334949*Sqrt[1 - 2*x])/64 + (98209*(1 - 2*x)^(3/2)
)/24 - (52011*(1 - 2*x)^(5/2))/64 + (11475*(1 - 2*x)^(7/2))/112 - (375*(1 - 2*x)^(9/2))/64

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Rubi [A]  time = 0.0170915, 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} \[ -\frac{375}{64} (1-2 x)^{9/2}+\frac{11475}{112} (1-2 x)^{7/2}-\frac{52011}{64} (1-2 x)^{5/2}+\frac{98209}{24} (1-2 x)^{3/2}-\frac{1334949}{64} \sqrt{1-2 x}-\frac{302379}{16 \sqrt{1-2 x}}+\frac{456533}{192 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

456533/(192*(1 - 2*x)^(3/2)) - 302379/(16*Sqrt[1 - 2*x]) - (1334949*Sqrt[1 - 2*x])/64 + (98209*(1 - 2*x)^(3/2)
)/24 - (52011*(1 - 2*x)^(5/2))/64 + (11475*(1 - 2*x)^(7/2))/112 - (375*(1 - 2*x)^(9/2))/64

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}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{456533}{64 (1-2 x)^{5/2}}-\frac{302379}{16 (1-2 x)^{3/2}}+\frac{1334949}{64 \sqrt{1-2 x}}-\frac{98209}{8} \sqrt{1-2 x}+\frac{260055}{64} (1-2 x)^{3/2}-\frac{11475}{16} (1-2 x)^{5/2}+\frac{3375}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac{456533}{192 (1-2 x)^{3/2}}-\frac{302379}{16 \sqrt{1-2 x}}-\frac{1334949}{64} \sqrt{1-2 x}+\frac{98209}{24} (1-2 x)^{3/2}-\frac{52011}{64} (1-2 x)^{5/2}+\frac{11475}{112} (1-2 x)^{7/2}-\frac{375}{64} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0178555, size = 43, normalized size = 0.47 \[ -\frac{7875 x^6+45225 x^5+130464 x^4+293785 x^3+1051833 x^2-2146758 x+714074}{21 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(714074 - 2146758*x + 1051833*x^2 + 293785*x^3 + 130464*x^4 + 45225*x^5 + 7875*x^6)/(21*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{7875\,{x}^{6}+45225\,{x}^{5}+130464\,{x}^{4}+293785\,{x}^{3}+1051833\,{x}^{2}-2146758\,x+714074}{21} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

-1/21*(7875*x^6+45225*x^5+130464*x^4+293785*x^3+1051833*x^2-2146758*x+714074)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.47299, size = 81, normalized size = 0.88 \begin{align*} -\frac{375}{64} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{11475}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{52011}{64} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{98209}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1334949}{64} \, \sqrt{-2 \, x + 1} + \frac{5929 \,{\left (1224 \, x - 535\right )}}{192 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

-375/64*(-2*x + 1)^(9/2) + 11475/112*(-2*x + 1)^(7/2) - 52011/64*(-2*x + 1)^(5/2) + 98209/24*(-2*x + 1)^(3/2)
- 1334949/64*sqrt(-2*x + 1) + 5929/192*(1224*x - 535)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.50993, size = 167, normalized size = 1.82 \begin{align*} -\frac{{\left (7875 \, x^{6} + 45225 \, x^{5} + 130464 \, x^{4} + 293785 \, x^{3} + 1051833 \, x^{2} - 2146758 \, x + 714074\right )} \sqrt{-2 \, x + 1}}{21 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \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)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(7875*x^6 + 45225*x^5 + 130464*x^4 + 293785*x^3 + 1051833*x^2 - 2146758*x + 714074)*sqrt(-2*x + 1)/(4*x^
2 - 4*x + 1)

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Sympy [A]  time = 24.6692, size = 82, normalized size = 0.89 \begin{align*} - \frac{375 \left (1 - 2 x\right )^{\frac{9}{2}}}{64} + \frac{11475 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} - \frac{52011 \left (1 - 2 x\right )^{\frac{5}{2}}}{64} + \frac{98209 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{1334949 \sqrt{1 - 2 x}}{64} - \frac{302379}{16 \sqrt{1 - 2 x}} + \frac{456533}{192 \left (1 - 2 x\right )^{\frac{3}{2}}} \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)**(5/2),x)

[Out]

-375*(1 - 2*x)**(9/2)/64 + 11475*(1 - 2*x)**(7/2)/112 - 52011*(1 - 2*x)**(5/2)/64 + 98209*(1 - 2*x)**(3/2)/24
- 1334949*sqrt(1 - 2*x)/64 - 302379/(16*sqrt(1 - 2*x)) + 456533/(192*(1 - 2*x)**(3/2))

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Giac [A]  time = 2.38325, size = 119, normalized size = 1.29 \begin{align*} -\frac{375}{64} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{11475}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{52011}{64} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{98209}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1334949}{64} \, \sqrt{-2 \, x + 1} - \frac{5929 \,{\left (1224 \, x - 535\right )}}{192 \,{\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

-375/64*(2*x - 1)^4*sqrt(-2*x + 1) - 11475/112*(2*x - 1)^3*sqrt(-2*x + 1) - 52011/64*(2*x - 1)^2*sqrt(-2*x + 1
) + 98209/24*(-2*x + 1)^(3/2) - 1334949/64*sqrt(-2*x + 1) - 5929/192*(1224*x - 535)/((2*x - 1)*sqrt(-2*x + 1))