3.21.69 \(\int \frac {(2+3 x)^7 (3+5 x)}{(1-2 x)^{3/2}} \, dx\) [2069]

Optimal. Leaf size=118 \[ \frac {9058973}{256 \sqrt {1-2 x}}+\frac {15647317}{128} \sqrt {1-2 x}-\frac {7882483}{128} (1-2 x)^{3/2}+\frac {4084101}{128} (1-2 x)^{5/2}-\frac {787185}{64} (1-2 x)^{7/2}+\frac {422919}{128} (1-2 x)^{9/2}-\frac {821583 (1-2 x)^{11/2}}{1408}+\frac {101331 (1-2 x)^{13/2}}{1664}-\frac {729}{256} (1-2 x)^{15/2} \]

[Out]

-7882483/128*(1-2*x)^(3/2)+4084101/128*(1-2*x)^(5/2)-787185/64*(1-2*x)^(7/2)+422919/128*(1-2*x)^(9/2)-821583/1
408*(1-2*x)^(11/2)+101331/1664*(1-2*x)^(13/2)-729/256*(1-2*x)^(15/2)+9058973/256/(1-2*x)^(1/2)[phone]/128*(1
-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 118, 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 {729}{256} (1-2 x)^{15/2}+\frac {101331 (1-2 x)^{13/2}}{1664}-\frac {821583 (1-2 x)^{11/2}}{1408}+\frac {422919}{128} (1-2 x)^{9/2}-\frac {787185}{64} (1-2 x)^{7/2}+\frac {4084101}{128} (1-2 x)^{5/2}-\frac {7882483}{128} (1-2 x)^{3/2}+\frac {15647317}{128} \sqrt {1-2 x}+\frac {9058973}{256 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^7*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

9058973/(256*Sqrt[1 - 2*x]) + (15647317*Sqrt[1 - 2*x])/128 - (7882483*(1 - 2*x)^(3/2))/128 + (4084101*(1 - 2*x
)^(5/2))/128 - (787185*(1 - 2*x)^(7/2))/64 + (422919*(1 - 2*x)^(9/2))/128 - (821583*(1 - 2*x)^(11/2))/1408 + (
101331*(1 - 2*x)^(13/2))/1664 - (729*(1 - 2*x)^(15/2))/256

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)^7 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {9058973}{256 (1-2 x)^{3/2}}-\frac {15647317}{128 \sqrt {1-2 x}}+\frac {23647449}{128} \sqrt {1-2 x}-\frac {20420505}{128} (1-2 x)^{3/2}+\frac {5510295}{64} (1-2 x)^{5/2}-\frac {3806271}{128} (1-2 x)^{7/2}+\frac {821583}{128} (1-2 x)^{9/2}-\frac {101331}{128} (1-2 x)^{11/2}+\frac {10935}{256} (1-2 x)^{13/2}\right ) \, dx\\ &=\frac {9058973}{256 \sqrt {1-2 x}}+\frac {15647317}{128} \sqrt {1-2 x}-\frac {7882483}{128} (1-2 x)^{3/2}+\frac {4084101}{128} (1-2 x)^{5/2}-\frac {787185}{64} (1-2 x)^{7/2}+\frac {422919}{128} (1-2 x)^{9/2}-\frac {821583 (1-2 x)^{11/2}}{1408}+\frac {101331 (1-2 x)^{13/2}}{1664}-\frac {729}{256} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.45 \begin {gather*} \frac {16936240-16881328 x-8106616 x^2-6921432 x^3-5949090 x^4-4220622 x^5-2168775 x^6-697653 x^7-104247 x^8}{143 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^7*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

(16936240 - 16881328*x - 8106616*x^2 - 6921432*x^3 - 5949090*x^4 - 4220622*x^5 - 2168775*x^6 - 697653*x^7 - 10
4247*x^8)/(143*Sqrt[1 - 2*x])

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Maple [A]
time = 0.14, size = 83, normalized size = 0.70

method	result	size
gosper	\(-\frac {104247 x^{8}+697653 x^{7}+2168775 x^{6}+4220622 x^{5}+5949090 x^{4}+6921432 x^{3}+8106616 x^{2}[phone] x -16936240}{143 \sqrt {1-2 x}}\)	\(50\)
risch	\(-\frac {104247 x^{8}+697653 x^{7}+2168775 x^{6}+4220622 x^{5}+5949090 x^{4}+6921432 x^{3}+8106616 x^{2}[phone] x -16936240}{143 \sqrt {1-2 x}}\)	\(50\)
trager	\(\frac {\left (104247 x^{8}+697653 x^{7}+2168775 x^{6}+4220622 x^{5}+5949090 x^{4}+6921432 x^{3}+8106616 x^{2}[phone] x -16936240\right ) \sqrt {1-2 x}}{-143+286 x}\)	\(57\)
derivativedivides	\(-\frac {7882483 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {4084101 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {787185 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {422919 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {821583 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {101331 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {729 \left (1-2 x \right )^{\frac {15}{2}}}{256}+\frac {9058973}{256 \sqrt {1-2 x}}+\frac {15647317 \sqrt {1-2 x}}{128}\)	\(83\)
default	\(-\frac {7882483 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {4084101 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {787185 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {422919 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {821583 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {101331 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {729 \left (1-2 x \right )^{\frac {15}{2}}}{256}+\frac {9058973}{256 \sqrt {1-2 x}}+\frac {15647317 \sqrt {1-2 x}}{128}\)	\(83\)
meijerg	\(-\frac {384 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-4672 \sqrt {\pi }+\frac {584 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {6216 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-30240 \sqrt {\pi }+\frac {945 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{4 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {17955 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{2 \sqrt {\pi }}+\frac {-22176 \sqrt {\pi }+\frac {693 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{32 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {66339 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{32 \sqrt {\pi }}+\frac {-\frac {307152 \sqrt {\pi }}{143}+\frac {19197 \sqrt {\pi }\, \left (-67584 x^{7}-43008 x^{6}-28672 x^{5}-20480 x^{4}-16384 x^{3}-16384 x^{2}-32768 x +32768\right )}{292864 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {10935 \left (\frac {32768 \sqrt {\pi }}{6435}-\frac {\sqrt {\pi }\, \left (-219648 x^{8}-135168 x^{7}-86016 x^{6}-57344 x^{5}-40960 x^{4}-32768 x^{3}-32768 x^{2}-65536 x +65536\right )}{12870 \sqrt {1-2 x}}\right )}{256 \sqrt {\pi }}\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^7*(3+5*x)/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-7882483/128*(1-2*x)^(3/2)+4084101/128*(1-2*x)^(5/2)-787185/64*(1-2*x)^(7/2)+422919/128*(1-2*x)^(9/2)-821583/1
408*(1-2*x)^(11/2)+101331/1664*(1-2*x)^(13/2)-729/256*(1-2*x)^(15/2)+9058973/256/(1-2*x)^(1/2)[phone]/128*(1
-2*x)^(1/2)

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Maxima [A]
time = 0.29, size = 82, normalized size = 0.69 \begin {gather*} -\frac {729}{256} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {101331}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {821583}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {422919}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {787185}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {4084101}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {7882483}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {15647317}{128} \, \sqrt {-2 \, x + 1} + \frac {9058973}{256 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-729/256*(-2*x + 1)^(15/2) + 101331/1664*(-2*x + 1)^(13/2) - 821583/1408*(-2*x + 1)^(11/2) + 422919/128*(-2*x
+ 1)^(9/2) - 787185/64*(-2*x + 1)^(7/2) + 4084101/128*(-2*x + 1)^(5/2) - 7882483/128*(-2*x + 1)^(3/2) + 156473
17/128*sqrt(-2*x + 1) + 9058973/256/sqrt(-2*x + 1)

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Fricas [A]
time = 0.81, size = 56, normalized size = 0.47 \begin {gather*} \frac {{\left (104247 \, x^{8} + 697653 \, x^{7} + 2168775 \, x^{6} + 4220622 \, x^{5} + 5949090 \, x^{4} + 6921432 \, x^{3} + 8106616 \, x^{2} + 16881328 \, x - 16936240\right )} \sqrt {-2 \, x + 1}}{143 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/143*(104247*x^8 + 697653*x^7 + 2168775*x^6 + 4220622*x^5 + 5949090*x^4 + 6921432*x^3 + 8106616*x^2 + 1688132
8*x - 16936240)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]
time = 35.53, size = 105, normalized size = 0.89 \begin {gather*} - \frac {729 \left (1 - 2 x\right )^{\frac {15}{2}}}{256} + \frac {101331 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {821583 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {422919 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {787185 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {4084101 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} - \frac {7882483 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} + \frac {15647317 \sqrt {1 - 2 x}}{128} + \frac {9058973}{256 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**7*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

-729*(1 - 2*x)**(15/2)/256 + 101331*(1 - 2*x)**(13/2)/1664 - 821583*(1 - 2*x)**(11/2)/1408 + 422919*(1 - 2*x)*
*(9/2)/128 - 787185*(1 - 2*x)**(7/2)/64 + 4084101*(1 - 2*x)**(5/2)/128 - 7882483*(1 - 2*x)**(3/2)/128 + 156473
17*sqrt(1 - 2*x)/128 + 9058973/(256*sqrt(1 - 2*x))

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Giac [A]
time = 1.47, size = 124, normalized size = 1.05 \begin {gather*} \frac {729}{256} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {101331}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {821583}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {422919}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {787185}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {4084101}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {7882483}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {15647317}{128} \, \sqrt {-2 \, x + 1} + \frac {9058973}{256 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

729/256*(2*x - 1)^7*sqrt(-2*x + 1) + 101331/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 821583/1408*(2*x - 1)^5*sqrt(-2*
x + 1) + 422919/128*(2*x - 1)^4*sqrt(-2*x + 1) + 787185/64*(2*x - 1)^3*sqrt(-2*x + 1) + 4084101/128*(2*x - 1)^
2*sqrt(-2*x + 1) - 7882483/128*(-2*x + 1)^(3/2) + 15647317/128*sqrt(-2*x + 1) + 9058973/256/sqrt(-2*x + 1)

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Mupad [B]
time = 1.21, size = 82, normalized size = 0.69 \begin {gather*} \frac {9058973}{256\,\sqrt {1-2\,x}}+\frac {15647317\,\sqrt {1-2\,x}}{128}-\frac {7882483\,{\left (1-2\,x\right )}^{3/2}}{128}+\frac {4084101\,{\left (1-2\,x\right )}^{5/2}}{128}-\frac {787185\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {422919\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {821583\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {101331\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {729\,{\left (1-2\,x\right )}^{15/2}}{256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^7*(5*x + 3))/(1 - 2*x)^(3/2),x)

[Out]

9058973/(256*(1 - 2*x)^(1/2)) + (15647317*(1 - 2*x)^(1/2))/128 - (7882483*(1 - 2*x)^(3/2))/128 + (4084101*(1 -
2*x)^(5/2))/128 - (787185*(1 - 2*x)^(7/2))/64 + (422919*(1 - 2*x)^(9/2))/128 - (821583*(1 - 2*x)^(11/2))/1408
+ (101331*(1 - 2*x)^(13/2))/1664 - (729*(1 - 2*x)^(15/2))/256

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