3.19.62 \(\int (1-2 x)^{3/2} (2+3 x) (3+5 x) \, dx\) [1862]

Optimal. Leaf size=40 \[ -\frac {77}{20} (1-2 x)^{5/2}+\frac {17}{7} (1-2 x)^{7/2}-\frac {5}{12} (1-2 x)^{9/2} \]

[Out]

-77/20*(1-2*x)^(5/2)+17/7*(1-2*x)^(7/2)-5/12*(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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {5}{12} (1-2 x)^{9/2}+\frac {17}{7} (1-2 x)^{7/2}-\frac {77}{20} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-77*(1 - 2*x)^(5/2))/20 + (17*(1 - 2*x)^(7/2))/7 - (5*(1 - 2*x)^(9/2))/12

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x) \, dx &=\int \left (\frac {77}{4} (1-2 x)^{3/2}-17 (1-2 x)^{5/2}+\frac {15}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {77}{20} (1-2 x)^{5/2}+\frac {17}{7} (1-2 x)^{7/2}-\frac {5}{12} (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}{105} (1-2 x)^{5/2} \left (193+335 x+175 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*((1 - 2*x)^(5/2)*(193 + 335*x + 175*x^2))

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

method	result	size
gosper	\(-\frac {\left (175 x^{2}+335 x +193\right ) \left (1-2 x \right )^{\frac {5}{2}}}{105}\)	\(20\)
derivativedivides	\(-\frac {77 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {17 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{12}\)	\(29\)
default	\(-\frac {77 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {17 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{12}\)	\(29\)
trager	\(\left (-\frac {20}{3} x^{4}-\frac {128}{21} x^{3}+\frac {131}{35} x^{2}+\frac {437}{105} x -\frac {193}{105}\right ) \sqrt {1-2 x}\)	\(29\)
risch	\(\frac {\left (700 x^{4}+640 x^{3}-393 x^{2}-437 x +193\right ) \left (-1+2 x \right )}{105 \sqrt {1-2 x}}\)	\(35\)
meijerg	\(-\frac {9 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {19 \sqrt {\pi }}{35}-\frac {19 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{280}}{\sqrt {\pi }}-\frac {45 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-77/20*(1-2*x)^(5/2)+17/7*(1-2*x)^(7/2)-5/12*(1-2*x)^(9/2)

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Maxima [A]
time = 0.29, size = 28, normalized size = 0.70 \begin {gather*} -\frac {5}{12} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {17}{7} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {77}{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)*(2+3*x)*(3+5*x),x, algorithm="maxima")

[Out]

-5/12*(-2*x + 1)^(9/2) + 17/7*(-2*x + 1)^(7/2) - 77/20*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.15, size = 29, normalized size = 0.72 \begin {gather*} -\frac {1}{105} \, {\left (700 \, x^{4} + 640 \, x^{3} - 393 \, x^{2} - 437 \, x + 193\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(700*x^4 + 640*x^3 - 393*x^2 - 437*x + 193)*sqrt(-2*x + 1)

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Sympy [A]
time = 3.67, size = 34, normalized size = 0.85 \begin {gather*} - \frac {5 \left (1 - 2 x\right )^{\frac {9}{2}}}{12} + \frac {17 \left (1 - 2 x\right )^{\frac {7}{2}}}{7} - \frac {77 \left (1 - 2 x\right )^{\frac {5}{2}}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*(1 - 2*x)**(9/2)/12 + 17*(1 - 2*x)**(7/2)/7 - 77*(1 - 2*x)**(5/2)/20

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Giac [A]
time = 1.35, size = 49, normalized size = 1.22 \begin {gather*} -\frac {5}{12} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {17}{7} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {77}{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)*(2+3*x)*(3+5*x),x, algorithm="giac")

[Out]

-5/12*(2*x - 1)^4*sqrt(-2*x + 1) - 17/7*(2*x - 1)^3*sqrt(-2*x + 1) - 77/20*(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 (2040\,x+175\,{\left (2\,x-1\right )}^2+597\right )}{420} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(2040*x + 175*(2*x - 1)^2 + 597))/420

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