3.22.51 \(\int \frac {(3+5 x)^2}{(1-2 x)^{5/2}} \, dx\) [2151]

Optimal. Leaf size=40 \[ \frac {121}{12 (1-2 x)^{3/2}}-\frac {55}{2 \sqrt {1-2 x}}-\frac {25}{4} \sqrt {1-2 x} \]

[Out]

121/12/(1-2*x)^(3/2)-55/2/(1-2*x)^(1/2)-25/4*(1-2*x)^(1/2)

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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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} -\frac {25}{4} \sqrt {1-2 x}-\frac {55}{2 \sqrt {1-2 x}}+\frac {121}{12 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/(1 - 2*x)^(5/2),x]

[Out]

121/(12*(1 - 2*x)^(3/2)) - 55/(2*Sqrt[1 - 2*x]) - (25*Sqrt[1 - 2*x])/4

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

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} -\frac {71-240 x+75 x^2}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(71 - 240*x + 75*x^2)/(1 - 2*x)^(3/2)

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Maple [A]
time = 0.17, size = 29, normalized size = 0.72

method	result	size
gosper	\(-\frac {75 x^{2}-240 x +71}{3 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(20\)
trager	\(-\frac {\left (75 x^{2}-240 x +71\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\)	\(27\)
risch	\(\frac {75 x^{2}-240 x +71}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(27\)
derivativedivides	\(\frac {121}{12 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {55}{2 \sqrt {1-2 x}}-\frac {25 \sqrt {1-2 x}}{4}\)	\(29\)
default	\(\frac {121}{12 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {55}{2 \sqrt {1-2 x}}-\frac {25 \sqrt {1-2 x}}{4}\)	\(29\)
meijerg	\(-\frac {6 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {10 \sqrt {\pi }-\frac {5 \sqrt {\pi }\, \left (-24 x +8\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {25 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{6 \sqrt {\pi }}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/12/(1-2*x)^(3/2)-55/2/(1-2*x)^(1/2)-25/4*(1-2*x)^(1/2)

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Maxima [A]
time = 0.37, size = 24, normalized size = 0.60 \begin {gather*} -\frac {25}{4} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (60 \, x - 19\right )}}{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)^(5/2),x, algorithm="maxima")

[Out]

-25/4*sqrt(-2*x + 1) + 11/12*(60*x - 19)/(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.78, size = 31, normalized size = 0.78 \begin {gather*} -\frac {{\left (75 \, x^{2} - 240 \, x + 71\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(75*x^2 - 240*x + 71)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]
time = 0.21, size = 65, normalized size = 1.62 \begin {gather*} - \frac {75 x^{2} \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} + \frac {240 x \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} - \frac {71 \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-75*x**2*sqrt(1 - 2*x)/(12*x**2 - 12*x + 3) + 240*x*sqrt(1 - 2*x)/(12*x**2 - 12*x + 3) - 71*sqrt(1 - 2*x)/(12*
x**2 - 12*x + 3)

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Giac [A]
time = 1.82, size = 31, normalized size = 0.78 \begin {gather*} -\frac {25}{4} \, \sqrt {-2 \, x + 1} - \frac {11 \, {\left (60 \, x - 19\right )}}{12 \, {\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

-25/4*sqrt(-2*x + 1) - 11/12*(60*x - 19)/((2*x - 1)*sqrt(-2*x + 1))

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Mupad [B]
time = 0.03, size = 29, normalized size = 0.72 \begin {gather*} \frac {75\,{\left (2\,x-1\right )}^2-660\,x+209}{\sqrt {1-2\,x}\,\left (24\,x-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^2/(1 - 2*x)^(5/2),x)

[Out]

(75*(2*x - 1)^2 - 660*x + 209)/((1 - 2*x)^(1/2)*(24*x - 12))

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