3.19.63 \(\int (1-2 x)^{3/2} (3+5 x) \, dx\) [1863]

Optimal. Leaf size=27 \[ -\frac {11}{10} (1-2 x)^{5/2}+\frac {5}{14} (1-2 x)^{7/2} \]

[Out]

-11/10*(1-2*x)^(5/2)+5/14*(1-2*x)^(7/2)

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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}{14} (1-2 x)^{7/2}-\frac {11}{10} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x),x]

[Out]

(-11*(1 - 2*x)^(5/2))/10 + (5*(1 - 2*x)^(7/2))/14

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) \, dx &=\int \left (\frac {11}{2} (1-2 x)^{3/2}-\frac {5}{2} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac {11}{10} (1-2 x)^{5/2}+\frac {5}{14} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{35} (1-2 x)^{5/2} (26+25 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/35*((1 - 2*x)^(5/2)*(26 + 25*x))

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Maple [A]
time = 0.10, size = 20, normalized size = 0.74

method	result	size
gosper	\(-\frac {\left (25 x +26\right ) \left (1-2 x \right )^{\frac {5}{2}}}{35}\)	\(15\)
derivativedivides	\(-\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{10}+\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{14}\)	\(20\)
default	\(-\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{10}+\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{14}\)	\(20\)
trager	\(\left (-\frac {20}{7} x^{3}-\frac {4}{35} x^{2}+\frac {79}{35} x -\frac {26}{35}\right ) \sqrt {1-2 x}\)	\(24\)
risch	\(\frac {\left (100 x^{3}+4 x^{2}-79 x +26\right ) \left (-1+2 x \right )}{35 \sqrt {1-2 x}}\)	\(30\)
meijerg	\(-\frac {9 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {\sqrt {\pi }}{7}-\frac {\sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{56}}{\sqrt {\pi }}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/10*(1-2*x)^(5/2)+5/14*(1-2*x)^(7/2)

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Maxima [A]
time = 0.27, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {11}{10} \, {\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),x, algorithm="maxima")

[Out]

5/14*(-2*x + 1)^(7/2) - 11/10*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.17, size = 24, normalized size = 0.89 \begin {gather*} -\frac {1}{35} \, {\left (100 \, x^{3} + 4 \, x^{2} - 79 \, x + 26\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),x, algorithm="fricas")

[Out]

-1/35*(100*x^3 + 4*x^2 - 79*x + 26)*sqrt(-2*x + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
time = 0.10, size = 54, normalized size = 2.00 \begin {gather*} - \frac {20 x^{3} \sqrt {1 - 2 x}}{7} - \frac {4 x^{2} \sqrt {1 - 2 x}}{35} + \frac {79 x \sqrt {1 - 2 x}}{35} - \frac {26 \sqrt {1 - 2 x}}{35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*x**3*sqrt(1 - 2*x)/7 - 4*x**2*sqrt(1 - 2*x)/35 + 79*x*sqrt(1 - 2*x)/35 - 26*sqrt(1 - 2*x)/35

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Giac [A]
time = 1.06, size = 33, normalized size = 1.22 \begin {gather*} -\frac {5}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {11}{10} \, {\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),x, algorithm="giac")

[Out]

-5/14*(2*x - 1)^3*sqrt(-2*x + 1) - 11/10*(2*x - 1)^2*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 )}^{5/2}\,\left (50\,x+52\right )}{70} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(5*x + 3),x)

[Out]

-((1 - 2*x)^(5/2)*(50*x + 52))/70

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