3.19.7 \(\int \sqrt {1-2 x} (3+5 x)^2 \, dx\) [1807]

Optimal. Leaf size=40 \[ -\frac {121}{12} (1-2 x)^{3/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {25}{28} (1-2 x)^{7/2} \]

[Out]

-121/12*(1-2*x)^(3/2)+11/2*(1-2*x)^(5/2)-25/28*(1-2*x)^(7/2)

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Rubi [A]
time = 0.00, 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}{28} (1-2 x)^{7/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {121}{12} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(3/2))/12 + (11*(1 - 2*x)^(5/2))/2 - (25*(1 - 2*x)^(7/2))/28

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 \sqrt {1-2 x} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} \sqrt {1-2 x}-\frac {55}{2} (1-2 x)^{3/2}+\frac {25}{4} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac {121}{12} (1-2 x)^{3/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {25}{28} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} -\frac {1}{21} (1-2 x)^{3/2} \left (115+156 x+75 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*((1 - 2*x)^(3/2)*(115 + 156*x + 75*x^2))

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

method	result	size
gosper	\(-\frac {\left (75 x^{2}+156 x +115\right ) \left (1-2 x \right )^{\frac {3}{2}}}{21}\)	\(20\)
trager	\(\left (\frac {50}{7} x^{3}+\frac {79}{7} x^{2}+\frac {74}{21} x -\frac {115}{21}\right ) \sqrt {1-2 x}\)	\(24\)
derivativedivides	\(-\frac {121 \left (1-2 x \right )^{\frac {3}{2}}}{12}+\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{2}-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{28}\)	\(29\)
default	\(-\frac {121 \left (1-2 x \right )^{\frac {3}{2}}}{12}+\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{2}-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{28}\)	\(29\)
risch	\(-\frac {\left (150 x^{3}+237 x^{2}+74 x -115\right ) \left (-1+2 x \right )}{21 \sqrt {1-2 x}}\)	\(30\)
meijerg	\(\frac {3 \sqrt {\pi }-\frac {3 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{2}}{\sqrt {\pi }}-\frac {15 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {10 \sqrt {\pi }}{21}-\frac {5 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{84}}{\sqrt {\pi }}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/12*(1-2*x)^(3/2)+11/2*(1-2*x)^(5/2)-25/28*(1-2*x)^(7/2)

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Maxima [A]
time = 0.27, size = 28, normalized size = 0.70 \begin {gather*} -\frac {25}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {11}{2} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/28*(-2*x + 1)^(7/2) + 11/2*(-2*x + 1)^(5/2) - 121/12*(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.03, size = 24, normalized size = 0.60 \begin {gather*} \frac {1}{21} \, {\left (150 \, x^{3} + 237 \, x^{2} + 74 \, x - 115\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(150*x^3 + 237*x^2 + 74*x - 115)*sqrt(-2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.71, size = 185, normalized size = 4.62 \begin {gather*} \begin {cases} \frac {10 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{7} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{35} - \frac {242 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \sqrt {10 x - 5}}{2625} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {10 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{7} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{35} - \frac {242 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x}}{2625} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((10*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/7 - 11*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/35 - 242*sqrt
(5)*I*(x + 3/5)*sqrt(10*x - 5)/525 - 2662*sqrt(5)*I*sqrt(10*x - 5)/2625, Abs(x + 3/5) > 11/10), (10*sqrt(5)*sq
rt(5 - 10*x)*(x + 3/5)**3/7 - 11*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/35 - 242*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)
/525 - 2662*sqrt(5)*sqrt(5 - 10*x)/2625, True))

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Giac [A]
time = 0.52, size = 42, normalized size = 1.05 \begin {gather*} \frac {25}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {11}{2} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/28*(2*x - 1)^3*sqrt(-2*x + 1) + 11/2*(2*x - 1)^2*sqrt(-2*x + 1) - 121/12*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {{\left (1-2\,x\right )}^{3/2}\,\left (924\,x+75\,{\left (2\,x-1\right )}^2+385\right )}{84} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(5*x + 3)^2,x)

[Out]

-((1 - 2*x)^(3/2)*(924*x + 75*(2*x - 1)^2 + 385))/84

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