3.21.25 \(\int \frac {(2+3 x)^3 (3+5 x)^3}{\sqrt {1-2 x}} \, dx\) [2025]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {90} \begin {gather*} -\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} \end {gather*}

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 90

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*}

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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-2 x} \left (18228666+17147586 x+20586249 x^2+20766885 x^3+14921900 x^4+6544125 x^5+1299375 x^6\right )}{5005} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5005*(Sqrt[1 - 2*x]*(18228666 + 17147586*x + 20586249*x^2 + 20766885*x^3 + 14921900*x^4 + 6544125*x^5 + 129
9375*x^6))

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Maple [A]
time = 0.12, size = 65, normalized size = 0.71

method	result	size
trager	\(\left (-\frac {3375}{13} x^{6}-\frac {186975}{143} x^{5}-\frac {426340}{143} x^{4}-\frac {4153377}{1001} x^{3}-\frac {20586249}{5005} x^{2}-\frac {17147586}{5005} x -\frac {18228666}{5005}\right ) \sqrt {1-2 x}\)	\(39\)
gosper	\(-\frac {\left (1299375 x^{6}+6544125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}}{5005}\)	\(40\)
risch	\(\frac {\left (-1+2 x \right ) \left (1299375 x^{6}+6544125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{5005 \sqrt {1-2 x}}\)	\(45\)
derivativedivides	\(\frac {100793 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {1334949 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {98209 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {28895 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {11475 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {3375 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {456533 \sqrt {1-2 x}}{64}\)	\(65\)
default	\(\frac {100793 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {1334949 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {98209 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {28895 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {11475 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {3375 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {456533 \sqrt {1-2 x}}{64}\)	\(65\)
meijerg	\(-\frac {108 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {684 \sqrt {\pi }-\frac {171 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{2}}{\sqrt {\pi }}-\frac {4059 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {34238 \sqrt {\pi }}{35}-\frac {17119 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{2240}}{\sqrt {\pi }}-\frac {20295 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {11400 \sqrt {\pi }}{77}-\frac {1425 \sqrt {\pi }\, \left (8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{9856}}{\sqrt {\pi }}-\frac {3375 \left (-\frac {2048 \sqrt {\pi }}{3003}+\frac {\sqrt {\pi }\, \left (29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{3003}\right )}{128 \sqrt {\pi }}\)	\(268\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 64, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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 = 1.29, size = 39, normalized size = 0.42 \begin {gather*} -\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 {gather*}

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 = 39.74, size = 82, normalized size = 0.89 \begin {gather*} - \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 {gather*}

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 = 0.85, size = 99, normalized size = 1.08 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {100793\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {456533\,\sqrt {1-2\,x}}{64}-\frac {1334949\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {98209\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {28895\,{\left (1-2\,x\right )}^{9/2}}{64}+\frac {11475\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {3375\,{\left (1-2\,x\right )}^{13/2}}{832} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(100793*(1 - 2*x)^(3/2))/16 - (456533*(1 - 2*x)^(1/2))/64 - (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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