3.21.12 \(\int \frac {(2+3 x)^5 (3+5 x)^2}{\sqrt {1-2 x}} \, dx\) [2012]

Optimal. Leaf size=105 \[ -\frac {2033647}{128} \sqrt {1-2 x}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {405}{128} (1-2 x)^{15/2} \]

[Out]

6206585/384*(1-2*x)^(3/2)-1623419/128*(1-2*x)^(5/2)+842415/128*(1-2*x)^(7/2)-285565/128*(1-2*x)^(9/2)+672003/1
408*(1-2*x)^(11/2)-97605/1664*(1-2*x)^(13/2)+405/128*(1-2*x)^(15/2)-2033647/128*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 105, 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 {405}{128} (1-2 x)^{15/2}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {2033647}{128} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2*
x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(1 - 2*x)^(15/2))/128

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)^5 (3+5 x)^2}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {2033647}{128 \sqrt {1-2 x}}-\frac {6206585}{128} \sqrt {1-2 x}+\frac {8117095}{128} (1-2 x)^{3/2}-\frac {5896905}{128} (1-2 x)^{5/2}+\frac {2570085}{128} (1-2 x)^{7/2}-\frac {672003}{128} (1-2 x)^{9/2}+\frac {97605}{128} (1-2 x)^{11/2}-\frac {6075}{128} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {2033647}{128} \sqrt {1-2 x}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {405}{128} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {1}{429} \sqrt {1-2 x} \left (3275704+3152152 x+4058988 x^2+4694340 x^3+4212525 x^4+2632743 x^5+1002375 x^6+173745 x^7\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/429*(Sqrt[1 - 2*x]*(3275704 + 3152152*x + 4058988*x^2 + 4694340*x^3 + 4212525*x^4 + 2632743*x^5 + 1002375*x
^6 + 173745*x^7))

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Maple [A]
time = 0.12, size = 74, normalized size = 0.70

method	result	size
trager	\(\left (-405 x^{7}-\frac {30375}{13} x^{6}-\frac {877581}{143} x^{5}-\frac {1404175}{143} x^{4}-\frac {1564780}{143} x^{3}-\frac {1352996}{143} x^{2}-\frac {3152152}{429} x -\frac {3275704}{429}\right ) \sqrt {1-2 x}\)	\(44\)
gosper	\(-\frac {\left (173745 x^{7}+1002375 x^{6}+2632743 x^{5}+4212525 x^{4}+4694340 x^{3}+4058988 x^{2}+3152152 x +3275704\right ) \sqrt {1-2 x}}{429}\)	\(45\)
risch	\(\frac {\left (-1+2 x \right ) \left (173745 x^{7}+1002375 x^{6}+2632743 x^{5}+4212525 x^{4}+4694340 x^{3}+4058988 x^{2}+3152152 x +3275704\right )}{429 \sqrt {1-2 x}}\)	\(50\)
derivativedivides	\(\frac {6206585 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {1623419 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {842415 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {285565 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {97605 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {405 \left (1-2 x \right )^{\frac {15}{2}}}{128}-\frac {2033647 \sqrt {1-2 x}}{128}\)	\(74\)
default	\(\frac {6206585 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {1623419 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {842415 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {285565 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {97605 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {405 \left (1-2 x \right )^{\frac {15}{2}}}{128}-\frac {2033647 \sqrt {1-2 x}}{128}\)	\(74\)
meijerg	\(-\frac {144 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {1040 \sqrt {\pi }-130 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {1810 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {14928 \sqrt {\pi }}{7}-\frac {933 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{56}}{\sqrt {\pi }}-\frac {28845 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{16 \sqrt {\pi }}+\frac {\frac {6792 \sqrt {\pi }}{11}-\frac {849 \sqrt {\pi }\, \left (8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{1408}}{\sqrt {\pi }}-\frac {6885 \left (-\frac {2048 \sqrt {\pi }}{3003}+\frac {\sqrt {\pi }\, \left (29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{3003}\right )}{32 \sqrt {\pi }}+\frac {\frac {2160 \sqrt {\pi }}{143}-\frac {135 \sqrt {\pi }\, \left (878592 x^{7}+473088 x^{6}+258048 x^{5}+143360 x^{4}+81920 x^{3}+49152 x^{2}+32768 x +32768\right ) \sqrt {1-2 x}}{292864}}{\sqrt {\pi }}\)	\(326\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6206585/384*(1-2*x)^(3/2)-1623419/128*(1-2*x)^(5/2)+842415/128*(1-2*x)^(7/2)-285565/128*(1-2*x)^(9/2)+672003/1
408*(1-2*x)^(11/2)-97605/1664*(1-2*x)^(13/2)+405/128*(1-2*x)^(15/2)-2033647/128*(1-2*x)^(1/2)

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Maxima [A]
time = 0.30, size = 73, normalized size = 0.70 \begin {gather*} \frac {405}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {97605}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {672003}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {285565}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {842415}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1623419}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {6206585}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2033647}{128} \, \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)^2/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

405/128*(-2*x + 1)^(15/2) - 97605/1664*(-2*x + 1)^(13/2) + 672003/1408*(-2*x + 1)^(11/2) - 285565/128*(-2*x +
1)^(9/2) + 842415/128*(-2*x + 1)^(7/2) - 1623419/128*(-2*x + 1)^(5/2) + 6206585/384*(-2*x + 1)^(3/2) - 2033647
/128*sqrt(-2*x + 1)

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Fricas [A]
time = 0.91, size = 44, normalized size = 0.42 \begin {gather*} -\frac {1}{429} \, {\left (173745 \, x^{7} + 1002375 \, x^{6} + 2632743 \, x^{5} + 4212525 \, x^{4} + 4694340 \, x^{3} + 4058988 \, x^{2} + 3152152 \, x + 3275704\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)^2/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/429*(173745*x^7 + 1002375*x^6 + 2632743*x^5 + 4212525*x^4 + 4694340*x^3 + 4058988*x^2 + 3152152*x + 3275704
)*sqrt(-2*x + 1)

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Sympy [A]
time = 48.76, size = 94, normalized size = 0.90 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} - \frac {97605 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} + \frac {672003 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {285565 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {842415 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} - \frac {1623419 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} + \frac {6206585 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} - \frac {2033647 \sqrt {1 - 2 x}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405*(1 - 2*x)**(15/2)/128 - 97605*(1 - 2*x)**(13/2)/1664 + 672003*(1 - 2*x)**(11/2)/1408 - 285565*(1 - 2*x)**(
9/2)/128 + 842415*(1 - 2*x)**(7/2)/128 - 1623419*(1 - 2*x)**(5/2)/128 + 6206585*(1 - 2*x)**(3/2)/384 - 2033647
*sqrt(1 - 2*x)/128

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Giac [A]
time = 1.58, size = 115, normalized size = 1.10 \begin {gather*} -\frac {405}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {97605}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {672003}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {285565}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {842415}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1623419}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {6206585}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2033647}{128} \, \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)^2/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-405/128*(2*x - 1)^7*sqrt(-2*x + 1) - 97605/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 672003/1408*(2*x - 1)^5*sqrt(-2*
x + 1) - 285565/128*(2*x - 1)^4*sqrt(-2*x + 1) - 842415/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1623419/128*(2*x - 1)
^2*sqrt(-2*x + 1) + 6206585/384*(-2*x + 1)^(3/2) - 2033647/128*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {6206585\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {2033647\,\sqrt {1-2\,x}}{128}-\frac {1623419\,{\left (1-2\,x\right )}^{5/2}}{128}+\frac {842415\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {285565\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {672003\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {97605\,{\left (1-2\,x\right )}^{13/2}}{1664}+\frac {405\,{\left (1-2\,x\right )}^{15/2}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6206585*(1 - 2*x)^(3/2))/384 - (2033647*(1 - 2*x)^(1/2))/128 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2
*x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(1 - 2*x)^(15/2))/128

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