3.19.74 \(\int (1-2 x)^{3/2} (3+5 x)^2 \, dx\) [1874]

Optimal. Leaf size=40 \[ -\frac {121}{20} (1-2 x)^{5/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {25}{36} (1-2 x)^{9/2} \]

[Out]

-121/20*(1-2*x)^(5/2)+55/14*(1-2*x)^(7/2)-25/36*(1-2*x)^(9/2)

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} -\frac {25}{36} (1-2 x)^{9/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {121}{20} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

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*} \int (1-2 x)^{3/2} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} (1-2 x)^{3/2}-\frac {55}{2} (1-2 x)^{5/2}+\frac {25}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {121}{20} (1-2 x)^{5/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {25}{36} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} -\frac {1}{315} (1-2 x)^{5/2} \left (887+1600 x+875 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*((1 - 2*x)^(5/2)*(887 + 1600*x + 875*x^2))

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Maple [A]
time = 0.10, size = 29, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (875 x^{2}+1600 x +887\right ) \left (1-2 x \right )^{\frac {5}{2}}}{315}\)	\(20\)
derivativedivides	\(-\frac {121 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {55 \left (1-2 x \right )^{\frac {7}{2}}}{14}-\frac {25 \left (1-2 x \right )^{\frac {9}{2}}}{36}\)	\(29\)
default	\(-\frac {121 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {55 \left (1-2 x \right )^{\frac {7}{2}}}{14}-\frac {25 \left (1-2 x \right )^{\frac {9}{2}}}{36}\)	\(29\)
trager	\(\left (-\frac {100}{9} x^{4}-\frac {580}{63} x^{3}+\frac {659}{105} x^{2}+\frac {1948}{315} x -\frac {887}{315}\right ) \sqrt {1-2 x}\)	\(29\)
risch	\(\frac {\left (3500 x^{4}+2900 x^{3}-1977 x^{2}-1948 x +887\right ) \left (-1+2 x \right )}{315 \sqrt {1-2 x}}\)	\(35\)
meijerg	\(-\frac {27 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {6 \sqrt {\pi }}{7}-\frac {3 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{28}}{\sqrt {\pi }}-\frac {75 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/20*(1-2*x)^(5/2)+55/14*(1-2*x)^(7/2)-25/36*(1-2*x)^(9/2)

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Maxima [A]
time = 0.26, size = 28, normalized size = 0.70 \begin {gather*} -\frac {25}{36} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {55}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {121}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(-2*x + 1)^(9/2) + 55/14*(-2*x + 1)^(7/2) - 121/20*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.20, size = 29, normalized size = 0.72 \begin {gather*} -\frac {1}{315} \, {\left (3500 \, x^{4} + 2900 \, x^{3} - 1977 \, x^{2} - 1948 \, x + 887\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(3500*x^4 + 2900*x^3 - 1977*x^2 - 1948*x + 887)*sqrt(-2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.97, size = 235, normalized size = 5.88 \begin {gather*} \begin {cases} - \frac {20 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{9} + \frac {220 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{63} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{7875} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{39375} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {20 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{9} + \frac {220 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{63} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{7875} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{39375} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-20*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/9 + 220*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/63 - 121*sq
rt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/525 - 2662*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/7875 - 29282*sqrt(5)*I*sqrt(
10*x - 5)/39375, Abs(x + 3/5) > 11/10), (-20*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/9 + 220*sqrt(5)*sqrt(5 - 10*x
)*(x + 3/5)**3/63 - 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/525 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/7875 -
29282*sqrt(5)*sqrt(5 - 10*x)/39375, True))

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Giac [A]
time = 1.30, size = 49, normalized size = 1.22 \begin {gather*} -\frac {25}{36} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {55}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {121}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(2*x - 1)^4*sqrt(-2*x + 1) - 55/14*(2*x - 1)^3*sqrt(-2*x + 1) - 121/20*(2*x - 1)^2*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {{\left (1-2\,x\right )}^{5/2}\,\left (9900\,x+875\,{\left (2\,x-1\right )}^2+2673\right )}{1260} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(5*x + 3)^2,x)

[Out]

-((1 - 2*x)^(5/2)*(9900*x + 875*(2*x - 1)^2 + 2673))/1260

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