3.22.59 \(\int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\) [2159]

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

[Out]

456533/192/(1-2*x)^(3/2)+98209/24*(1-2*x)^(3/2)-52011/64*(1-2*x)^(5/2)+11475/112*(1-2*x)^(7/2)-375/64*(1-2*x)^
(9/2)-302379/16/(1-2*x)^(1/2)-1334949/64*(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 {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}} \end {gather*}

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 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}{(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*}

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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {714074-2146758 x+1051833 x^2+293785 x^3+130464 x^4+45225 x^5+7875 x^6}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {7875 x^{6}+45225 x^{5}+130464 x^{4}+293785 x^{3}+1051833 x^{2}-2146758 x +714074}{21 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(40\)
trager	\(-\frac {\left (7875 x^{6}+45225 x^{5}+130464 x^{4}+293785 x^{3}+1051833 x^{2}-2146758 x +714074\right ) \sqrt {1-2 x}}{21 \left (-1+2 x \right )^{2}}\)	\(47\)
risch	\(\frac {7875 x^{6}+45225 x^{5}+130464 x^{4}+293785 x^{3}+1051833 x^{2}-2146758 x +714074}{21 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(47\)
derivativedivides	\(\frac {456533}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {98209 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {52011 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {11475 \left (1-2 x \right )^{\frac {7}{2}}}{112}-\frac {375 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {302379}{16 \sqrt {1-2 x}}-\frac {1334949 \sqrt {1-2 x}}{64}\)	\(65\)
default	\(\frac {456533}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {98209 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {52011 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {11475 \left (1-2 x \right )^{\frac {7}{2}}}{112}-\frac {375 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {302379}{16 \sqrt {1-2 x}}-\frac {1334949 \sqrt {1-2 x}}{64}\)	\(65\)
meijerg	\(-\frac {144 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {684 \sqrt {\pi }-\frac {171 \sqrt {\pi }\, \left (-24 x +8\right )}{2 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {1353 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {34238 \sqrt {\pi }}{3}-\frac {17119 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{192 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {6765 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {34200 \sqrt {\pi }}{7}-\frac {4275 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {1125 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}\)	\(266\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 60, normalized size = 0.65 \begin {gather*} -\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 {gather*}

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 = 0.98, size = 51, normalized size = 0.55 \begin {gather*} -\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 {gather*}

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 = 17.03, size = 82, normalized size = 0.89 \begin {gather*} - \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 {gather*}

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 = 1.75, size = 88, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {302379\,x}{8}-\frac {3172015}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {1334949\,\sqrt {1-2\,x}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {52011\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {11475\,{\left (1-2\,x\right )}^{7/2}}{112}-\frac {375\,{\left (1-2\,x\right )}^{9/2}}{64} \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)^(5/2),x)

[Out]

((302379*x)/8 - 3172015/192)/(1 - 2*x)^(3/2) - (1334949*(1 - 2*x)^(1/2))/64 + (98209*(1 - 2*x)^(3/2))/24 - (52
011*(1 - 2*x)^(5/2))/64 + (11475*(1 - 2*x)^(7/2))/112 - (375*(1 - 2*x)^(9/2))/64

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