3.1816 \(\int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx\)

Optimal. Leaf size=92 \[ -\frac{225}{64} (1-2 x)^{15/2}+\frac{11475}{208} (1-2 x)^{13/2}-\frac{260055}{704} (1-2 x)^{11/2}+\frac{98209}{72} (1-2 x)^{9/2}-\frac{190707}{64} (1-2 x)^{7/2}+\frac{302379}{80} (1-2 x)^{5/2}-\frac{456533}{192} (1-2 x)^{3/2} \]

[Out]

(-456533*(1 - 2*x)^(3/2))/192 + (302379*(1 - 2*x)^(5/2))/80 - (190707*(1 - 2*x)^(7/2))/64 + (98209*(1 - 2*x)^(
9/2))/72 - (260055*(1 - 2*x)^(11/2))/704 + (11475*(1 - 2*x)^(13/2))/208 - (225*(1 - 2*x)^(15/2))/64

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Rubi [A]  time = 0.0167199, 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} \[ -\frac{225}{64} (1-2 x)^{15/2}+\frac{11475}{208} (1-2 x)^{13/2}-\frac{260055}{704} (1-2 x)^{11/2}+\frac{98209}{72} (1-2 x)^{9/2}-\frac{190707}{64} (1-2 x)^{7/2}+\frac{302379}{80} (1-2 x)^{5/2}-\frac{456533}{192} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(3/2))/192 + (302379*(1 - 2*x)^(5/2))/80 - (190707*(1 - 2*x)^(7/2))/64 + (98209*(1 - 2*x)^(
9/2))/72 - (260055*(1 - 2*x)^(11/2))/704 + (11475*(1 - 2*x)^(13/2))/208 - (225*(1 - 2*x)^(15/2))/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 \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx &=\int \left (\frac{456533}{64} \sqrt{1-2 x}-\frac{302379}{16} (1-2 x)^{3/2}+\frac{1334949}{64} (1-2 x)^{5/2}-\frac{98209}{8} (1-2 x)^{7/2}+\frac{260055}{64} (1-2 x)^{9/2}-\frac{11475}{16} (1-2 x)^{11/2}+\frac{3375}{64} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac{456533}{192} (1-2 x)^{3/2}+\frac{302379}{80} (1-2 x)^{5/2}-\frac{190707}{64} (1-2 x)^{7/2}+\frac{98209}{72} (1-2 x)^{9/2}-\frac{260055}{704} (1-2 x)^{11/2}+\frac{11475}{208} (1-2 x)^{13/2}-\frac{225}{64} (1-2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0189037, size = 43, normalized size = 0.47 \[ -\frac{(1-2 x)^{3/2} \left (1447875 x^6+7016625 x^5+15061950 x^4+18934285 x^3+15577455 x^2+8871906 x+3420622\right )}{6435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(3420622 + 8871906*x + 15577455*x^2 + 18934285*x^3 + 15061950*x^4 + 7016625*x^5 + 1447875*x^
6))/6435

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Maple [A]  time = 0.004, size = 40, normalized size = 0.4 \begin{align*} -{\frac{1447875\,{x}^{6}+7016625\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}+8871906\,x+3420622}{6435} \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)^(1/2),x)

[Out]

-1/6435*(1447875*x^6+7016625*x^5+15061950*x^4+18934285*x^3+15577455*x^2+8871906*x+3420622)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.53866, size = 86, normalized size = 0.93 \begin{align*} -\frac{225}{64} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{11475}{208} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{260055}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{98209}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{190707}{64} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{302379}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{456533}{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)^(1/2),x, algorithm="maxima")

[Out]

-225/64*(-2*x + 1)^(15/2) + 11475/208*(-2*x + 1)^(13/2) - 260055/704*(-2*x + 1)^(11/2) + 98209/72*(-2*x + 1)^(
9/2) - 190707/64*(-2*x + 1)^(7/2) + 302379/80*(-2*x + 1)^(5/2) - 456533/192*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.58581, size = 180, normalized size = 1.96 \begin{align*} \frac{1}{6435} \,{\left (2895750 \, x^{7} + 12585375 \, x^{6} + 23107275 \, x^{5} + 22806620 \, x^{4} + 12220625 \, x^{3} + 2166357 \, x^{2} - 2030662 \, x - 3420622\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]

1/6435*(2895750*x^7 + 12585375*x^6 + 23107275*x^5 + 22806620*x^4 + 12220625*x^3 + 2166357*x^2 - 2030662*x - 34
20622)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.31466, size = 82, normalized size = 0.89 \begin{align*} - \frac{225 \left (1 - 2 x\right )^{\frac{15}{2}}}{64} + \frac{11475 \left (1 - 2 x\right )^{\frac{13}{2}}}{208} - \frac{260055 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{98209 \left (1 - 2 x\right )^{\frac{9}{2}}}{72} - \frac{190707 \left (1 - 2 x\right )^{\frac{7}{2}}}{64} + \frac{302379 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} - \frac{456533 \left (1 - 2 x\right )^{\frac{3}{2}}}{192} \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]

-225*(1 - 2*x)**(15/2)/64 + 11475*(1 - 2*x)**(13/2)/208 - 260055*(1 - 2*x)**(11/2)/704 + 98209*(1 - 2*x)**(9/2
)/72 - 190707*(1 - 2*x)**(7/2)/64 + 302379*(1 - 2*x)**(5/2)/80 - 456533*(1 - 2*x)**(3/2)/192

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Giac [A]  time = 2.59763, size = 143, normalized size = 1.55 \begin{align*} \frac{225}{64} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{11475}{208} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{260055}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{98209}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{190707}{64} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{302379}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{456533}{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)^(1/2),x, algorithm="giac")

[Out]

225/64*(2*x - 1)^7*sqrt(-2*x + 1) + 11475/208*(2*x - 1)^6*sqrt(-2*x + 1) + 260055/704*(2*x - 1)^5*sqrt(-2*x +
1) + 98209/72*(2*x - 1)^4*sqrt(-2*x + 1) + 190707/64*(2*x - 1)^3*sqrt(-2*x + 1) + 302379/80*(2*x - 1)^2*sqrt(-
2*x + 1) - 456533/192*(-2*x + 1)^(3/2)