3.21.1 \(\int \frac {(2+3 x)^5 (3+5 x)}{\sqrt {1-2 x}} \, dx\) [2001]

Optimal. Leaf size=92 \[ -\frac {184877}{64} \sqrt {1-2 x}+\frac {60025}{24} (1-2 x)^{3/2}-\frac {103929}{64} (1-2 x)^{5/2}+\frac {5355}{8} (1-2 x)^{7/2}-\frac {10815}{64} (1-2 x)^{9/2}+\frac {1053}{44} (1-2 x)^{11/2}-\frac {1215}{832} (1-2 x)^{13/2} \]

[Out]

60025/24*(1-2*x)^(3/2)-103929/64*(1-2*x)^(5/2)+5355/8*(1-2*x)^(7/2)-10815/64*(1-2*x)^(9/2)+1053/44*(1-2*x)^(11
/2)-1215/832*(1-2*x)^(13/2)-184877/64*(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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {1215}{832} (1-2 x)^{13/2}+\frac {1053}{44} (1-2 x)^{11/2}-\frac {10815}{64} (1-2 x)^{9/2}+\frac {5355}{8} (1-2 x)^{7/2}-\frac {103929}{64} (1-2 x)^{5/2}+\frac {60025}{24} (1-2 x)^{3/2}-\frac {184877}{64} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-184877*Sqrt[1 - 2*x])/64 + (60025*(1 - 2*x)^(3/2))/24 - (103929*(1 - 2*x)^(5/2))/64 + (5355*(1 - 2*x)^(7/2))
/8 - (10815*(1 - 2*x)^(9/2))/64 + (1053*(1 - 2*x)^(11/2))/44 - (1215*(1 - 2*x)^(13/2))/832

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)^5 (3+5 x)}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {184877}{64 \sqrt {1-2 x}}-\frac {60025}{8} \sqrt {1-2 x}+\frac {519645}{64} (1-2 x)^{3/2}-\frac {37485}{8} (1-2 x)^{5/2}+\frac {97335}{64} (1-2 x)^{7/2}-\frac {1053}{4} (1-2 x)^{9/2}+\frac {1215}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {184877}{64} \sqrt {1-2 x}+\frac {60025}{24} (1-2 x)^{3/2}-\frac {103929}{64} (1-2 x)^{5/2}+\frac {5355}{8} (1-2 x)^{7/2}-\frac {10815}{64} (1-2 x)^{9/2}+\frac {1053}{44} (1-2 x)^{11/2}-\frac {1215}{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 {1}{429} \sqrt {1-2 x} \left (638648+597464 x+707436 x^2+698580 x^3+488925 x^4+208251 x^5+40095 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/429*(Sqrt[1 - 2*x]*(638648 + 597464*x + 707436*x^2 + 698580*x^3 + 488925*x^4 + 208251*x^5 + 40095*x^6))

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Maple [A]
time = 0.14, size = 65, normalized size = 0.71

method	result	size
trager	\(\left (-\frac {1215}{13} x^{6}-\frac {69417}{143} x^{5}-\frac {162975}{143} x^{4}-\frac {232860}{143} x^{3}-\frac {235812}{143} x^{2}-\frac {597464}{429} x -\frac {638648}{429}\right ) \sqrt {1-2 x}\)	\(39\)
gosper	\(-\frac {\left (40095 x^{6}+208251 x^{5}+488925 x^{4}+698580 x^{3}+707436 x^{2}+597464 x +638648\right ) \sqrt {1-2 x}}{429}\)	\(40\)
risch	\(\frac {\left (-1+2 x \right ) \left (40095 x^{6}+208251 x^{5}+488925 x^{4}+698580 x^{3}+707436 x^{2}+597464 x +638648\right )}{429 \sqrt {1-2 x}}\)	\(45\)
derivativedivides	\(\frac {60025 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {103929 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {5355 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {10815 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {11}{2}}}{44}-\frac {1215 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {184877 \sqrt {1-2 x}}{64}\)	\(65\)
default	\(\frac {60025 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {103929 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {5355 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {10815 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {11}{2}}}{44}-\frac {1215 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {184877 \sqrt {1-2 x}}{64}\)	\(65\)
meijerg	\(-\frac {48 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {880 \sqrt {\pi }}{3}-\frac {110 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{3}}{\sqrt {\pi }}-\frac {420 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {2736 \sqrt {\pi }}{7}-\frac {171 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{56}}{\sqrt {\pi }}-\frac {3915 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{16 \sqrt {\pi }}+\frac {\frac {4248 \sqrt {\pi }}{77}-\frac {531 \sqrt {\pi }\, \left (8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{9856}}{\sqrt {\pi }}-\frac {1215 \left (-\frac {2048 \sqrt {\pi }}{3003}+\frac {\sqrt {\pi }\, \left (29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{3003}\right )}{128 \sqrt {\pi }}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

60025/24*(1-2*x)^(3/2)-103929/64*(1-2*x)^(5/2)+5355/8*(1-2*x)^(7/2)-10815/64*(1-2*x)^(9/2)+1053/44*(1-2*x)^(11
/2)-1215/832*(1-2*x)^(13/2)-184877/64*(1-2*x)^(1/2)

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Maxima [A]
time = 0.30, size = 64, normalized size = 0.70 \begin {gather*} -\frac {1215}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {1053}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {10815}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {5355}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {103929}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {60025}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {184877}{64} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-1215/832*(-2*x + 1)^(13/2) + 1053/44*(-2*x + 1)^(11/2) - 10815/64*(-2*x + 1)^(9/2) + 5355/8*(-2*x + 1)^(7/2)
- 103929/64*(-2*x + 1)^(5/2) + 60025/24*(-2*x + 1)^(3/2) - 184877/64*sqrt(-2*x + 1)

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Fricas [A]
time = 0.87, size = 39, normalized size = 0.42 \begin {gather*} -\frac {1}{429} \, {\left (40095 \, x^{6} + 208251 \, x^{5} + 488925 \, x^{4} + 698580 \, x^{3} + 707436 \, x^{2} + 597464 \, x + 638648\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/429*(40095*x^6 + 208251*x^5 + 488925*x^4 + 698580*x^3 + 707436*x^2 + 597464*x + 638648)*sqrt(-2*x + 1)

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Sympy [A]
time = 40.31, size = 82, normalized size = 0.89 \begin {gather*} - \frac {1215 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {1053 \left (1 - 2 x\right )^{\frac {11}{2}}}{44} - \frac {10815 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {5355 \left (1 - 2 x\right )^{\frac {7}{2}}}{8} - \frac {103929 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {60025 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {184877 \sqrt {1 - 2 x}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215*(1 - 2*x)**(13/2)/832 + 1053*(1 - 2*x)**(11/2)/44 - 10815*(1 - 2*x)**(9/2)/64 + 5355*(1 - 2*x)**(7/2)/8
- 103929*(1 - 2*x)**(5/2)/64 + 60025*(1 - 2*x)**(3/2)/24 - 184877*sqrt(1 - 2*x)/64

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Giac [A]
time = 0.63, size = 99, normalized size = 1.08 \begin {gather*} -\frac {1215}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {1053}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {10815}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {5355}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {103929}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {60025}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {184877}{64} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1215/832*(2*x - 1)^6*sqrt(-2*x + 1) - 1053/44*(2*x - 1)^5*sqrt(-2*x + 1) - 10815/64*(2*x - 1)^4*sqrt(-2*x + 1
) - 5355/8*(2*x - 1)^3*sqrt(-2*x + 1) - 103929/64*(2*x - 1)^2*sqrt(-2*x + 1) + 60025/24*(-2*x + 1)^(3/2) - 184
877/64*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {60025\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {184877\,\sqrt {1-2\,x}}{64}-\frac {103929\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {5355\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {10815\,{\left (1-2\,x\right )}^{9/2}}{64}+\frac {1053\,{\left (1-2\,x\right )}^{11/2}}{44}-\frac {1215\,{\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)^5*(5*x + 3))/(1 - 2*x)^(1/2),x)

[Out]

(60025*(1 - 2*x)^(3/2))/24 - (184877*(1 - 2*x)^(1/2))/64 - (103929*(1 - 2*x)^(5/2))/64 + (5355*(1 - 2*x)^(7/2)
)/8 - (10815*(1 - 2*x)^(9/2))/64 + (1053*(1 - 2*x)^(11/2))/44 - (1215*(1 - 2*x)^(13/2))/832

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