3.21.75 \(\int \frac {(2+3 x) (3+5 x)}{(1-2 x)^{3/2}} \, dx\) [2075]

Optimal. Leaf size=38 \[ \frac {77}{4 \sqrt {1-2 x}}+17 \sqrt {1-2 x}-\frac {5}{4} (1-2 x)^{3/2} \]

[Out]

-5/4*(1-2*x)^(3/2)+77/4/(1-2*x)^(1/2)+17*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 38, 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}{4} (1-2 x)^{3/2}+17 \sqrt {1-2 x}+\frac {77}{4 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

77/(4*Sqrt[1 - 2*x]) + 17*Sqrt[1 - 2*x] - (5*(1 - 2*x)^(3/2))/4

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 \frac {(2+3 x) (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {77}{4 (1-2 x)^{3/2}}-\frac {17}{\sqrt {1-2 x}}+\frac {15}{4} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {77}{4 \sqrt {1-2 x}}+17 \sqrt {1-2 x}-\frac {5}{4} (1-2 x)^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.53 \begin {gather*} \frac {35-29 x-5 x^2}{\sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35 - 29*x - 5*x^2)/Sqrt[1 - 2*x]

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Maple [A]
time = 0.12, size = 29, normalized size = 0.76

method	result	size
gosper	\(-\frac {5 x^{2}+29 x -35}{\sqrt {1-2 x}}\)	\(20\)
risch	\(-\frac {5 x^{2}+29 x -35}{\sqrt {1-2 x}}\)	\(20\)
trager	\(\frac {\left (5 x^{2}+29 x -35\right ) \sqrt {1-2 x}}{-1+2 x}\)	\(26\)
derivativedivides	\(-\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {77}{4 \sqrt {1-2 x}}+17 \sqrt {1-2 x}\)	\(29\)
default	\(-\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {77}{4 \sqrt {1-2 x}}+17 \sqrt {1-2 x}\)	\(29\)
meijerg	\(-\frac {6 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-19 \sqrt {\pi }+\frac {19 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {15 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(1-2*x)^(3/2)+77/4/(1-2*x)^(1/2)+17*(1-2*x)^(1/2)

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Maxima [A]
time = 0.30, size = 28, normalized size = 0.74 \begin {gather*} -\frac {5}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 17 \, \sqrt {-2 \, x + 1} + \frac {77}{4 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)

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Fricas [A]
time = 1.08, size = 25, normalized size = 0.66 \begin {gather*} \frac {{\left (5 \, x^{2} + 29 \, x - 35\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*x^2 + 29*x - 35)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]
time = 5.57, size = 32, normalized size = 0.84 \begin {gather*} - \frac {5 \left (1 - 2 x\right )^{\frac {3}{2}}}{4} + 17 \sqrt {1 - 2 x} + \frac {77}{4 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*(1 - 2*x)**(3/2)/4 + 17*sqrt(1 - 2*x) + 77/(4*sqrt(1 - 2*x))

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Giac [A]
time = 1.08, size = 28, normalized size = 0.74 \begin {gather*} -\frac {5}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 17 \, \sqrt {-2 \, x + 1} + \frac {77}{4 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.61 \begin {gather*} -\frac {136\,x+5\,{\left (2\,x-1\right )}^2-145}{4\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(136*x + 5*(2*x - 1)^2 - 145)/(4*(1 - 2*x)^(1/2))

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