3.21.94 \(\int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\) [2094]

Optimal. Leaf size=105 \[ \frac {3195731}{128 \sqrt {1-2 x}}+\frac {9836211}{128} \sqrt {1-2 x}-\frac {4324397}{128} (1-2 x)^{3/2}+\frac {9504551}{640} (1-2 x)^{5/2}-\frac {4177401}{896} (1-2 x)^{7/2}+\frac {122385}{128} (1-2 x)^{9/2}-\frac {161325 (1-2 x)^{11/2}}{1408}+\frac {10125 (1-2 x)^{13/2}}{1664} \]

[Out]

-4324397/128*(1-2*x)^(3/2)+9504551/640*(1-2*x)^(5/2)-4177401/896*(1-2*x)^(7/2)+122385/128*(1-2*x)^(9/2)-161325
/1408*(1-2*x)^(11/2)+10125/1664*(1-2*x)^(13/2)+3195731/128/(1-2*x)^(1/2)+9836211/128*(1-2*x)^(1/2)

________________________________________________________________________________________

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 {10125 (1-2 x)^{13/2}}{1664}-\frac {161325 (1-2 x)^{11/2}}{1408}+\frac {122385}{128} (1-2 x)^{9/2}-\frac {4177401}{896} (1-2 x)^{7/2}+\frac {9504551}{640} (1-2 x)^{5/2}-\frac {4324397}{128} (1-2 x)^{3/2}+\frac {9836211}{128} \sqrt {1-2 x}+\frac {3195731}{128 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3195731/(128*Sqrt[1 - 2*x]) + (9836211*Sqrt[1 - 2*x])/128 - (4324397*(1 - 2*x)^(3/2))/128 + (9504551*(1 - 2*x)
^(5/2))/640 - (4177401*(1 - 2*x)^(7/2))/896 + (122385*(1 - 2*x)^(9/2))/128 - (161325*(1 - 2*x)^(11/2))/1408 +
(10125*(1 - 2*x)^(13/2))/1664

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)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {3195731}{128 (1-2 x)^{3/2}}-\frac {9836211}{128 \sqrt {1-2 x}}+\frac {12973191}{128} \sqrt {1-2 x}-\frac {9504551}{128} (1-2 x)^{3/2}+\frac {4177401}{128} (1-2 x)^{5/2}-\frac {1101465}{128} (1-2 x)^{7/2}+\frac {161325}{128} (1-2 x)^{9/2}-\frac {10125}{128} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac {3195731}{128 \sqrt {1-2 x}}+\frac {9836211}{128} \sqrt {1-2 x}-\frac {4324397}{128} (1-2 x)^{3/2}+\frac {9504551}{640} (1-2 x)^{5/2}-\frac {4177401}{896} (1-2 x)^{7/2}+\frac {122385}{128} (1-2 x)^{9/2}-\frac {161325 (1-2 x)^{11/2}}{1408}+\frac {10125 (1-2 x)^{13/2}}{1664}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} \frac {395714912-393552752 x-184884496 x^2-147527176 x^3-111095730 x^4-63495075 x^5-23058000 x^6-3898125 x^7}{5005 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(395714912 - 393552752*x - 184884496*x^2 - 147527176*x^3 - 111095730*x^4 - 63495075*x^5 - 23058000*x^6 - 38981
25*x^7)/(5005*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 74, normalized size = 0.70

method	result	size
gosper	\(-\frac {3898125 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x -395714912}{5005 \sqrt {1-2 x}}\)	\(45\)
risch	\(-\frac {3898125 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x -395714912}{5005 \sqrt {1-2 x}}\)	\(45\)
trager	\(\frac {\left (3898125 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x -395714912\right ) \sqrt {1-2 x}}{-5005+10010 x}\)	\(52\)
derivativedivides	\(-\frac {4324397 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {4177401 \left (1-2 x \right )^{\frac {7}{2}}}{896}+\frac {122385 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {161325 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {10125 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {3195731}{128 \sqrt {1-2 x}}+\frac {9836211 \sqrt {1-2 x}}{128}\)	\(74\)
default	\(-\frac {4324397 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {4177401 \left (1-2 x \right )^{\frac {7}{2}}}{896}+\frac {122385 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {161325 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {10125 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {3195731}{128 \sqrt {1-2 x}}+\frac {9836211 \sqrt {1-2 x}}{128}\)	\(74\)
meijerg	\(-\frac {432 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-4752 \sqrt {\pi }+\frac {594 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {5598 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-\frac {117184 \sqrt {\pi }}{5}+\frac {1831 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{10 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {91947 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {76920 \sqrt {\pi }}{7}+\frac {9615 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {45225 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}+\frac {-\frac {54000 \sqrt {\pi }}{143}+\frac {3375 \sqrt {\pi }\, \left (-67584 x^{7}-43008 x^{6}-28672 x^{5}-20480 x^{4}-16384 x^{3}-16384 x^{2}-32768 x +32768\right )}{292864 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(324\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4324397/128*(1-2*x)^(3/2)+9504551/640*(1-2*x)^(5/2)-4177401/896*(1-2*x)^(7/2)+122385/128*(1-2*x)^(9/2)-161325
/1408*(1-2*x)^(11/2)+10125/1664*(1-2*x)^(13/2)+3195731/128/(1-2*x)^(1/2)+9836211/128*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 73, normalized size = 0.70 \begin {gather*} \frac {10125}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {161325}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {122385}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {4177401}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {9504551}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {4324397}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {9836211}{128} \, \sqrt {-2 \, x + 1} + \frac {3195731}{128 \, \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)^3/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

10125/1664*(-2*x + 1)^(13/2) - 161325/1408*(-2*x + 1)^(11/2) + 122385/128*(-2*x + 1)^(9/2) - 4177401/896*(-2*x
+ 1)^(7/2) + 9504551/640*(-2*x + 1)^(5/2) - 4324397/128*(-2*x + 1)^(3/2) + 9836211/128*sqrt(-2*x + 1) + 31957
31/128/sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.71, size = 51, normalized size = 0.49 \begin {gather*} \frac {{\left (3898125 \, x^{7} + 23058000 \, x^{6} + 63495075 \, x^{5} + 111095730 \, x^{4} + 147527176 \, x^{3} + 184884496 \, x^{2} + 393552752 \, x - 395714912\right )} \sqrt {-2 \, x + 1}}{5005 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/5005*(3898125*x^7 + 23058000*x^6 + 63495075*x^5 + 111095730*x^4 + 147527176*x^3 + 184884496*x^2 + 393552752*
x - 395714912)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]
time = 29.05, size = 94, normalized size = 0.90 \begin {gather*} \frac {10125 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {161325 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {122385 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {4177401 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} + \frac {9504551 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} - \frac {4324397 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} + \frac {9836211 \sqrt {1 - 2 x}}{128} + \frac {3195731}{128 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10125*(1 - 2*x)**(13/2)/1664 - 161325*(1 - 2*x)**(11/2)/1408 + 122385*(1 - 2*x)**(9/2)/128 - 4177401*(1 - 2*x)
**(7/2)/896 + 9504551*(1 - 2*x)**(5/2)/640 - 4324397*(1 - 2*x)**(3/2)/128 + 9836211*sqrt(1 - 2*x)/128 + 319573
1/(128*sqrt(1 - 2*x))

________________________________________________________________________________________

Giac [A]
time = 1.43, size = 108, normalized size = 1.03 \begin {gather*} \frac {10125}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {161325}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {122385}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {4177401}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {9504551}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {4324397}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {9836211}{128} \, \sqrt {-2 \, x + 1} + \frac {3195731}{128 \, \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)^3/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

10125/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 161325/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 122385/128*(2*x - 1)^4*sqrt(-
2*x + 1) + 4177401/896*(2*x - 1)^3*sqrt(-2*x + 1) + 9504551/640*(2*x - 1)^2*sqrt(-2*x + 1) - 4324397/128*(-2*x
+ 1)^(3/2) + 9836211/128*sqrt(-2*x + 1) + 3195731/128/sqrt(-2*x + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 73, normalized size = 0.70 \begin {gather*} \frac {3195731}{128\,\sqrt {1-2\,x}}+\frac {9836211\,\sqrt {1-2\,x}}{128}-\frac {4324397\,{\left (1-2\,x\right )}^{3/2}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {4177401\,{\left (1-2\,x\right )}^{7/2}}{896}+\frac {122385\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {161325\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {10125\,{\left (1-2\,x\right )}^{13/2}}{1664} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3195731/(128*(1 - 2*x)^(1/2)) + (9836211*(1 - 2*x)^(1/2))/128 - (4324397*(1 - 2*x)^(3/2))/128 + (9504551*(1 -
2*x)^(5/2))/640 - (4177401*(1 - 2*x)^(7/2))/896 + (122385*(1 - 2*x)^(9/2))/128 - (161325*(1 - 2*x)^(11/2))/140
8 + (10125*(1 - 2*x)^(13/2))/1664

________________________________________________________________________________________