3.22.36 \(\int \frac {(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx\) [2136]

Optimal. Leaf size=79 \[ \frac {26411}{96 (1-2 x)^{3/2}}-\frac {57281}{32 \sqrt {1-2 x}}-\frac {24843}{16} \sqrt {1-2 x}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {405}{224} (1-2 x)^{7/2} \]

[Out]

26411/96/(1-2*x)^(3/2)+3591/16*(1-2*x)^(3/2)-4671/160*(1-2*x)^(5/2)+405/224*(1-2*x)^(7/2)-57281/32/(1-2*x)^(1/
2)-24843/16*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} \frac {405}{224} (1-2 x)^{7/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {24843}{16} \sqrt {1-2 x}-\frac {57281}{32 \sqrt {1-2 x}}+\frac {26411}{96 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

26411/(96*(1 - 2*x)^(3/2)) - 57281/(32*Sqrt[1 - 2*x]) - (24843*Sqrt[1 - 2*x])/16 + (3591*(1 - 2*x)^(3/2))/16 -
(4671*(1 - 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {26411}{32 (1-2 x)^{5/2}}-\frac {57281}{32 (1-2 x)^{3/2}}+\frac {24843}{16 \sqrt {1-2 x}}-\frac {10773}{16} \sqrt {1-2 x}+\frac {4671}{32} (1-2 x)^{3/2}-\frac {405}{32} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {26411}{96 (1-2 x)^{3/2}}-\frac {57281}{32 \sqrt {1-2 x}}-\frac {24843}{16} \sqrt {1-2 x}+\frac {3591}{16} (1-2 x)^{3/2}-\frac {4671}{160} (1-2 x)^{5/2}+\frac {405}{224} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {301408-909264 x+435312 x^2+105624 x^3+33858 x^4+6075 x^5}{105 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(301408 - 909264*x + 435312*x^2 + 105624*x^3 + 33858*x^4 + 6075*x^5)/(1 - 2*x)^(3/2)

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Maple [A]
time = 0.16, size = 56, normalized size = 0.71

method	result	size
gosper	\(-\frac {6075 x^{5}+33858 x^{4}+105624 x^{3}+435312 x^{2}-909264 x +301408}{105 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(35\)
trager	\(-\frac {\left (6075 x^{5}+33858 x^{4}+105624 x^{3}+435312 x^{2}-909264 x +301408\right ) \sqrt {1-2 x}}{105 \left (-1+2 x \right )^{2}}\)	\(42\)
risch	\(\frac {6075 x^{5}+33858 x^{4}+105624 x^{3}+435312 x^{2}-909264 x +301408}{105 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(42\)
derivativedivides	\(\frac {26411}{96 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {3591 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {4671 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {405 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {57281}{32 \sqrt {1-2 x}}-\frac {24843 \sqrt {1-2 x}}{16}\)	\(56\)
default	\(\frac {26411}{96 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {3591 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {4671 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {405 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {57281}{32 \sqrt {1-2 x}}-\frac {24843 \sqrt {1-2 x}}{16}\)	\(56\)
meijerg	\(-\frac {32 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {368 \sqrt {\pi }}{3}-\frac {46 \sqrt {\pi }\, \left (-24 x +8\right )}{3 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {188 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {1152 \sqrt {\pi }-\frac {9 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {441 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {1080 \sqrt {\pi }}{7}-\frac {135 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\)	\(213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

26411/96/(1-2*x)^(3/2)+3591/16*(1-2*x)^(3/2)-4671/160*(1-2*x)^(5/2)+405/224*(1-2*x)^(7/2)-57281/32/(1-2*x)^(1/
2)-24843/16*(1-2*x)^(1/2)

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Maxima [A]
time = 0.27, size = 51, normalized size = 0.65 \begin {gather*} \frac {405}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4671}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {3591}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {24843}{16} \, \sqrt {-2 \, x + 1} + \frac {343 \, {\left (501 \, x - 212\right )}}{48 \, {\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)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

405/224*(-2*x + 1)^(7/2) - 4671/160*(-2*x + 1)^(5/2) + 3591/16*(-2*x + 1)^(3/2) - 24843/16*sqrt(-2*x + 1) + 34
3/48*(501*x - 212)/(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.95, size = 46, normalized size = 0.58 \begin {gather*} -\frac {{\left (6075 \, x^{5} + 33858 \, x^{4} + 105624 \, x^{3} + 435312 \, x^{2} - 909264 \, x + 301408\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)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/105*(6075*x^5 + 33858*x^4 + 105624*x^3 + 435312*x^2 - 909264*x + 301408)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]
time = 15.30, size = 70, normalized size = 0.89 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {4671 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} + \frac {3591 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {24843 \sqrt {1 - 2 x}}{16} - \frac {57281}{32 \sqrt {1 - 2 x}} + \frac {26411}{96 \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)/(1-2*x)**(5/2),x)

[Out]

405*(1 - 2*x)**(7/2)/224 - 4671*(1 - 2*x)**(5/2)/160 + 3591*(1 - 2*x)**(3/2)/16 - 24843*sqrt(1 - 2*x)/16 - 572
81/(32*sqrt(1 - 2*x)) + 26411/(96*(1 - 2*x)**(3/2))

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Giac [A]
time = 2.57, size = 72, normalized size = 0.91 \begin {gather*} -\frac {405}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4671}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {3591}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {24843}{16} \, \sqrt {-2 \, x + 1} - \frac {343 \, {\left (501 \, x - 212\right )}}{48 \, {\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)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-405/224*(2*x - 1)^3*sqrt(-2*x + 1) - 4671/160*(2*x - 1)^2*sqrt(-2*x + 1) + 3591/16*(-2*x + 1)^(3/2) - 24843/1
6*sqrt(-2*x + 1) - 343/48*(501*x - 212)/((2*x - 1)*sqrt(-2*x + 1))

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Mupad [B]
time = 0.03, size = 50, normalized size = 0.63 \begin {gather*} \frac {\frac {57281\,x}{16}-\frac {18179}{12}}{{\left (1-2\,x\right )}^{3/2}}-\frac {24843\,\sqrt {1-2\,x}}{16}+\frac {3591\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {4671\,{\left (1-2\,x\right )}^{5/2}}{160}+\frac {405\,{\left (1-2\,x\right )}^{7/2}}{224} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((57281*x)/16 - 18179/12)/(1 - 2*x)^(3/2) - (24843*(1 - 2*x)^(1/2))/16 + (3591*(1 - 2*x)^(3/2))/16 - (4671*(1
- 2*x)^(5/2))/160 + (405*(1 - 2*x)^(7/2))/224

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