\(\int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\) [2085]

Optimal result

Integrand size = 24, antiderivative size = 79 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {41503}{32 \sqrt {1-2 x}}+\frac {91091}{32} \sqrt {1-2 x}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {75}{32} (1-2 x)^{9/2} \]

[Out]

-39977/48*(1-2*x)^(3/2)+17541/80*(1-2*x)^(5/2)-7695/224*(1-2*x)^(7/2)+75/32*(1-2*x)^(9/2)+41503/32/(1-2*x)^(1/
2)+91091/32*(1-2*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {75}{32} (1-2 x)^{9/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {91091}{32} \sqrt {1-2 x}+\frac {41503}{32 \sqrt {1-2 x}} \]

[In]

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

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))/80
- (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/32

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*} \text {integral}& = \int \left (\frac {41503}{32 (1-2 x)^{3/2}}-\frac {91091}{32 \sqrt {1-2 x}}+\frac {39977}{16} \sqrt {1-2 x}-\frac {17541}{16} (1-2 x)^{3/2}+\frac {7695}{32} (1-2 x)^{5/2}-\frac {675}{32} (1-2 x)^{7/2}\right ) \, dx \\ & = \frac {41503}{32 \sqrt {1-2 x}}+\frac {91091}{32} \sqrt {1-2 x}-\frac {39977}{48} (1-2 x)^{3/2}+\frac {17541}{80} (1-2 x)^{5/2}-\frac {7695}{224} (1-2 x)^{7/2}+\frac {75}{32} (1-2 x)^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {367286-359726 x-150253 x^2-88443 x^3-38025 x^4-7875 x^5}{105 \sqrt {1-2 x}} \]

[In]

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

[Out]

(367286 - 359726*x - 150253*x^2 - 88443*x^3 - 38025*x^4 - 7875*x^5)/(105*Sqrt[1 - 2*x])

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {7875 x^{5}+38025 x^{4}+88443 x^{3}+150253 x^{2}+359726 x -367286}{105 \sqrt {1-2 x}}\)	\(35\)
risch	\(-\frac {7875 x^{5}+38025 x^{4}+88443 x^{3}+150253 x^{2}+359726 x -367286}{105 \sqrt {1-2 x}}\)	\(35\)
pseudoelliptic	\(\frac {-7875 x^{5}-38025 x^{4}-88443 x^{3}-150253 x^{2}-359726 x +367286}{105 \sqrt {1-2 x}}\)	\(35\)
trager	\(\frac {\left (7875 x^{5}+38025 x^{4}+88443 x^{3}+150253 x^{2}+359726 x -367286\right ) \sqrt {1-2 x}}{-105+210 x}\)	\(42\)
derivativedivides	\(-\frac {39977 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {17541 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {7695 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {75 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {41503}{32 \sqrt {1-2 x}}+\frac {91091 \sqrt {1-2 x}}{32}\)	\(56\)
default	\(-\frac {39977 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {17541 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {7695 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {75 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {41503}{32 \sqrt {1-2 x}}+\frac {91091 \sqrt {1-2 x}}{32}\)	\(56\)
meijerg	\(-\frac {72 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-564 \sqrt {\pi }+\frac {141 \sqrt {\pi }\, \left (-8 x +8\right )}{2 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {883 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{2 \sqrt {\pi }}+\frac {-\frac {5526 \sqrt {\pi }}{5}+\frac {2763 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{320 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {135 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-\frac {600 \sqrt {\pi }}{7}+\frac {75 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(213\)

[In]

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

[Out]

-1/105/(1-2*x)^(1/2)*(7875*x^5+38025*x^4+88443*x^3+150253*x^2+359726*x-367286)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {{\left (7875 \, x^{5} + 38025 \, x^{4} + 88443 \, x^{3} + 150253 \, x^{2} + 359726 \, x - 367286\right )} \sqrt {-2 \, x + 1}}{105 \, {\left (2 \, x - 1\right )}} \]

[In]

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

[Out]

1/105*(7875*x^5 + 38025*x^4 + 88443*x^3 + 150253*x^2 + 359726*x - 367286)*sqrt(-2*x + 1)/(2*x - 1)

Sympy [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {75 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {7695 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {17541 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {39977 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} + \frac {91091 \sqrt {1 - 2 x}}{32} + \frac {41503}{32 \sqrt {1 - 2 x}} \]

[In]

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

[Out]

75*(1 - 2*x)**(9/2)/32 - 7695*(1 - 2*x)**(7/2)/224 + 17541*(1 - 2*x)**(5/2)/80 - 39977*(1 - 2*x)**(3/2)/48 + 9
1091*sqrt(1 - 2*x)/32 + 41503/(32*sqrt(1 - 2*x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {75}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {7695}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {17541}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {39977}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {91091}{32} \, \sqrt {-2 \, x + 1} + \frac {41503}{32 \, \sqrt {-2 \, x + 1}} \]

[In]

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

[Out]

75/32*(-2*x + 1)^(9/2) - 7695/224*(-2*x + 1)^(7/2) + 17541/80*(-2*x + 1)^(5/2) - 39977/48*(-2*x + 1)^(3/2) + 9
1091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {75}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {7695}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {17541}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {39977}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {91091}{32} \, \sqrt {-2 \, x + 1} + \frac {41503}{32 \, \sqrt {-2 \, x + 1}} \]

[In]

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

[Out]

75/32*(2*x - 1)^4*sqrt(-2*x + 1) + 7695/224*(2*x - 1)^3*sqrt(-2*x + 1) + 17541/80*(2*x - 1)^2*sqrt(-2*x + 1) -
39977/48*(-2*x + 1)^(3/2) + 91091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {41503}{32\,\sqrt {1-2\,x}}+\frac {91091\,\sqrt {1-2\,x}}{32}-\frac {39977\,{\left (1-2\,x\right )}^{3/2}}{48}+\frac {17541\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {7695\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {75\,{\left (1-2\,x\right )}^{9/2}}{32} \]

[In]

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

[Out]

41503/(32*(1 - 2*x)^(1/2)) + (91091*(1 - 2*x)^(1/2))/32 - (39977*(1 - 2*x)^(3/2))/48 + (17541*(1 - 2*x)^(5/2))
/80 - (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2))/32