\(\int (1-2 x)^{5/2} (3+5 x)^2 \, dx\) [1946]

Optimal result

Integrand size = 17, antiderivative size = 40 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {121}{28} (1-2 x)^{7/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {25}{44} (1-2 x)^{11/2} \]

[Out]

-121/28*(1-2*x)^(7/2)+55/18*(1-2*x)^(9/2)-25/44*(1-2*x)^(11/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {25}{44} (1-2 x)^{11/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {121}{28} (1-2 x)^{7/2} \]

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {121}{4} (1-2 x)^{5/2}-\frac {55}{2} (1-2 x)^{7/2}+\frac {25}{4} (1-2 x)^{9/2}\right ) \, dx \\ & = -\frac {121}{28} (1-2 x)^{7/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {25}{44} (1-2 x)^{11/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {1}{693} (1-2 x)^{7/2} \left (1271+2660 x+1575 x^2\right ) \]

[In]

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

[Out]

-1/693*((1 - 2*x)^(7/2)*(1271 + 2660*x + 1575*x^2))

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50

method	result	size
gosper	\(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (1575 x^{2}+2660 x +1271\right )}{693}\)	\(20\)
pseudoelliptic	\(\frac {\sqrt {1-2 x}\, \left (-1+2 x \right )^{3} \left (1575 x^{2}+2660 x +1271\right )}{693}\)	\(27\)
derivativedivides	\(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {55 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {25 \left (1-2 x \right )^{\frac {11}{2}}}{44}\)	\(29\)
default	\(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {55 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {25 \left (1-2 x \right )^{\frac {11}{2}}}{44}\)	\(29\)
trager	\(\left (\frac {200}{11} x^{5}+\frac {340}{99} x^{4}-\frac {12302}{693} x^{3}-\frac {289}{231} x^{2}+\frac {4966}{693} x -\frac {1271}{693}\right ) \sqrt {1-2 x}\)	\(34\)
risch	\(-\frac {\left (12600 x^{5}+2380 x^{4}-12302 x^{3}-867 x^{2}+4966 x -1271\right ) \left (-1+2 x \right )}{693 \sqrt {1-2 x}}\)	\(40\)
meijerg	\(\frac {\frac {9 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{14}}{\sqrt {\pi }}-\frac {225 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{16 \sqrt {\pi }}+\frac {\frac {50 \sqrt {\pi }}{693}-\frac {25 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{5544}}{\sqrt {\pi }}\)	\(131\)

[In]

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

[Out]

-1/693*(1-2*x)^(7/2)*(1575*x^2+2660*x+1271)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {1}{693} \, {\left (12600 \, x^{5} + 2380 \, x^{4} - 12302 \, x^{3} - 867 \, x^{2} + 4966 \, x - 1271\right )} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

1/693*(12600*x^5 + 2380*x^4 - 12302*x^3 - 867*x^2 + 4966*x - 1271)*sqrt(-2*x + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (34) = 68\).

Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {200 x^{5} \sqrt {1 - 2 x}}{11} + \frac {340 x^{4} \sqrt {1 - 2 x}}{99} - \frac {12302 x^{3} \sqrt {1 - 2 x}}{693} - \frac {289 x^{2} \sqrt {1 - 2 x}}{231} + \frac {4966 x \sqrt {1 - 2 x}}{693} - \frac {1271 \sqrt {1 - 2 x}}{693} \]

[In]

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

[Out]

200*x**5*sqrt(1 - 2*x)/11 + 340*x**4*sqrt(1 - 2*x)/99 - 12302*x**3*sqrt(1 - 2*x)/693 - 289*x**2*sqrt(1 - 2*x)/
231 + 4966*x*sqrt(1 - 2*x)/693 - 1271*sqrt(1 - 2*x)/693

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {25}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {55}{18} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {121}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]

[In]

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

[Out]

-25/44*(-2*x + 1)^(11/2) + 55/18*(-2*x + 1)^(9/2) - 121/28*(-2*x + 1)^(7/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.22 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {25}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {55}{18} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {121}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

25/44*(2*x - 1)^5*sqrt(-2*x + 1) + 55/18*(2*x - 1)^4*sqrt(-2*x + 1) + 121/28*(2*x - 1)^3*sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {{\left (1-2\,x\right )}^{7/2}\,\left (16940\,x+1575\,{\left (2\,x-1\right )}^2+3509\right )}{2772} \]

[In]

int((1 - 2*x)^(5/2)*(5*x + 3)^2,x)

[Out]

-((1 - 2*x)^(7/2)*(16940*x + 1575*(2*x - 1)^2 + 3509))/2772