3.22.47 \(\int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\) [2147]

Optimal. Leaf size=92 \[ \frac {290521}{192 (1-2 x)^{3/2}}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951}{64} \sqrt {1-2 x}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {225}{64} (1-2 x)^{9/2} \]

[Out]

290521/192/(1-2*x)^(3/2)+40453/16*(1-2*x)^(3/2)-159111/320*(1-2*x)^(5/2)+13905/224*(1-2*x)^(7/2)-225/64*(1-2*x
)^(9/2)-381073/32/(1-2*x)^(1/2)-832951/64*(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 {225}{64} (1-2 x)^{9/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {832951}{64} \sqrt {1-2 x}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {290521}{192 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(5/2),x]

[Out]

290521/(192*(1 - 2*x)^(3/2)) - 381073/(32*Sqrt[1 - 2*x]) - (832951*Sqrt[1 - 2*x])/64 + (40453*(1 - 2*x)^(3/2))
/16 - (159111*(1 - 2*x)^(5/2))/320 + (13905*(1 - 2*x)^(7/2))/224 - (225*(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)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {290521}{64 (1-2 x)^{5/2}}-\frac {381073}{32 (1-2 x)^{3/2}}+\frac {832951}{64 \sqrt {1-2 x}}-\frac {121359}{16} \sqrt {1-2 x}+\frac {159111}{64} (1-2 x)^{3/2}-\frac {13905}{32} (1-2 x)^{5/2}+\frac {2025}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {290521}{192 (1-2 x)^{3/2}}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951}{64} \sqrt {1-2 x}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {225}{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 {2238664-6731112 x+3294996 x^2+915492 x^3+402489 x^4+137700 x^5+23625 x^6}{105 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(5/2),x]

[Out]

-1/105*(2238664 - 6731112*x + 3294996*x^2 + 915492*x^3 + 402489*x^4 + 137700*x^5 + 23625*x^6)/(1 - 2*x)^(3/2)

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

method	result	size
gosper	\(-\frac {23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664}{105 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(40\)
trager	\(-\frac {\left (23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664\right ) \sqrt {1-2 x}}{105 \left (-1+2 x \right )^{2}}\)	\(47\)
risch	\(\frac {23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664}{105 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(47\)
derivativedivides	\(\frac {290521}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {40453 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {159111 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {225 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951 \sqrt {1-2 x}}{64}\)	\(65\)
default	\(\frac {290521}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {40453 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {159111 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {225 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951 \sqrt {1-2 x}}{64}\)	\(65\)
meijerg	\(-\frac {96 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {448 \sqrt {\pi }-\frac {56 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {2612 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {7216 \sqrt {\pi }-\frac {451 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {4203 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {20880 \sqrt {\pi }}{7}-\frac {1305 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{448 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {675 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4*(3+5*x)^2/(1-2*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

290521/192/(1-2*x)^(3/2)+40453/16*(1-2*x)^(3/2)-159111/320*(1-2*x)^(5/2)+13905/224*(1-2*x)^(7/2)-225/64*(1-2*x
)^(9/2)-381073/32/(1-2*x)^(1/2)-832951/64*(1-2*x)^(1/2)

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Maxima [A]
time = 0.40, size = 60, normalized size = 0.65 \begin {gather*} -\frac {225}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {13905}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {159111}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} + \frac {3773 \, {\left (1212 \, x - 529\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)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-225/64*(-2*x + 1)^(9/2) + 13905/224*(-2*x + 1)^(7/2) - 159111/320*(-2*x + 1)^(5/2) + 40453/16*(-2*x + 1)^(3/2
) - 832951/64*sqrt(-2*x + 1) + 3773/192*(1212*x - 529)/(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.12, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (23625 \, x^{6} + 137700 \, x^{5} + 402489 \, x^{4} + 915492 \, x^{3} + 3294996 \, x^{2} - 6731112 \, x + 2238664\right )} \sqrt {-2 \, x + 1}}{105 \, {\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)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/105*(23625*x^6 + 137700*x^5 + 402489*x^4 + 915492*x^3 + 3294996*x^2 - 6731112*x + 2238664)*sqrt(-2*x + 1)/(
4*x^2 - 4*x + 1)

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Sympy [A]
time = 18.28, size = 82, normalized size = 0.89 \begin {gather*} - \frac {225 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {13905 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {159111 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {40453 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {832951 \sqrt {1 - 2 x}}{64} - \frac {381073}{32 \sqrt {1 - 2 x}} + \frac {290521}{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)**4*(3+5*x)**2/(1-2*x)**(5/2),x)

[Out]

-225*(1 - 2*x)**(9/2)/64 + 13905*(1 - 2*x)**(7/2)/224 - 159111*(1 - 2*x)**(5/2)/320 + 40453*(1 - 2*x)**(3/2)/1
6 - 832951*sqrt(1 - 2*x)/64 - 381073/(32*sqrt(1 - 2*x)) + 290521/(192*(1 - 2*x)**(3/2))

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Giac [A]
time = 0.97, size = 88, normalized size = 0.96 \begin {gather*} -\frac {225}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {13905}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {159111}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} - \frac {3773 \, {\left (1212 \, x - 529\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)^4*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-225/64*(2*x - 1)^4*sqrt(-2*x + 1) - 13905/224*(2*x - 1)^3*sqrt(-2*x + 1) - 159111/320*(2*x - 1)^2*sqrt(-2*x +
1) + 40453/16*(-2*x + 1)^(3/2) - 832951/64*sqrt(-2*x + 1) - 3773/192*(1212*x - 529)/((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 {381073\,x}{16}-\frac {1995917}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {832951\,\sqrt {1-2\,x}}{64}+\frac {40453\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {159111\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {13905\,{\left (1-2\,x\right )}^{7/2}}{224}-\frac {225\,{\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)^4*(5*x + 3)^2)/(1 - 2*x)^(5/2),x)

[Out]

((381073*x)/16 - 1995917/192)/(1 - 2*x)^(3/2) - (832951*(1 - 2*x)^(1/2))/64 + (40453*(1 - 2*x)^(3/2))/16 - (15
9111*(1 - 2*x)^(5/2))/320 + (13905*(1 - 2*x)^(7/2))/224 - (225*(1 - 2*x)^(9/2))/64

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