3.20.34 \(\int (1-2 x)^{5/2} (3+5 x) \, dx\) [1934]

Optimal. Leaf size=27 \[ -\frac {11}{14} (1-2 x)^{7/2}+\frac {5}{18} (1-2 x)^{9/2} \]

[Out]

-11/14*(1-2*x)^(7/2)+5/18*(1-2*x)^(9/2)

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {5}{18} (1-2 x)^{9/2}-\frac {11}{14} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x),x]

[Out]

(-11*(1 - 2*x)^(7/2))/14 + (5*(1 - 2*x)^(9/2))/18

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)^{5/2} (3+5 x) \, dx &=\int \left (\frac {11}{2} (1-2 x)^{5/2}-\frac {5}{2} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {11}{14} (1-2 x)^{7/2}+\frac {5}{18} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{63} (1-2 x)^{7/2} (32+35 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/63*((1 - 2*x)^(7/2)*(32 + 35*x))

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Maple [A]
time = 0.11, size = 20, normalized size = 0.74

method	result	size
gosper	\(-\frac {\left (35 x +32\right ) \left (1-2 x \right )^{\frac {7}{2}}}{63}\)	\(15\)
derivativedivides	\(-\frac {11 \left (1-2 x \right )^{\frac {7}{2}}}{14}+\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{18}\)	\(20\)
default	\(-\frac {11 \left (1-2 x \right )^{\frac {7}{2}}}{14}+\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{18}\)	\(20\)
trager	\(\left (\frac {40}{9} x^{4}-\frac {164}{63} x^{3}-\frac {58}{21} x^{2}+\frac {157}{63} x -\frac {32}{63}\right ) \sqrt {1-2 x}\)	\(29\)
risch	\(-\frac {\left (280 x^{4}-164 x^{3}-174 x^{2}+157 x -32\right ) \left (-1+2 x \right )}{63 \sqrt {1-2 x}}\)	\(35\)
meijerg	\(\frac {\frac {3 \sqrt {\pi }}{7}-\frac {3 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{14}}{\sqrt {\pi }}-\frac {75 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/14*(1-2*x)^(7/2)+5/18*(1-2*x)^(9/2)

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Maxima [A]
time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{18} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {11}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/18*(-2*x + 1)^(9/2) - 11/14*(-2*x + 1)^(7/2)

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Fricas [A]
time = 1.32, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{63} \, {\left (280 \, x^{4} - 164 \, x^{3} - 174 \, x^{2} + 157 \, x - 32\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/63*(280*x^4 - 164*x^3 - 174*x^2 + 157*x - 32)*sqrt(-2*x + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
time = 0.20, size = 70, normalized size = 2.59 \begin {gather*} \frac {40 x^{4} \sqrt {1 - 2 x}}{9} - \frac {164 x^{3} \sqrt {1 - 2 x}}{63} - \frac {58 x^{2} \sqrt {1 - 2 x}}{21} + \frac {157 x \sqrt {1 - 2 x}}{63} - \frac {32 \sqrt {1 - 2 x}}{63} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40*x**4*sqrt(1 - 2*x)/9 - 164*x**3*sqrt(1 - 2*x)/63 - 58*x**2*sqrt(1 - 2*x)/21 + 157*x*sqrt(1 - 2*x)/63 - 32*s
qrt(1 - 2*x)/63

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Giac [A]
time = 0.60, size = 33, normalized size = 1.22 \begin {gather*} \frac {5}{18} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {11}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/18*(2*x - 1)^4*sqrt(-2*x + 1) + 11/14*(2*x - 1)^3*sqrt(-2*x + 1)

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.52 \begin {gather*} -\frac {{\left (1-2\,x\right )}^{7/2}\,\left (70\,x+64\right )}{126} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)*(5*x + 3),x)

[Out]

-((1 - 2*x)^(7/2)*(70*x + 64))/126

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