3.21.97 \(\int \frac {(2+3 x) (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\) [2097]

Optimal. Leaf size=66 \[ \frac {9317}{16 \sqrt {1-2 x}}+\frac {8349}{8} \sqrt {1-2 x}-\frac {935}{4} (1-2 x)^{3/2}+\frac {335}{8} (1-2 x)^{5/2}-\frac {375}{112} (1-2 x)^{7/2} \]

[Out]

-935/4*(1-2*x)^(3/2)+335/8*(1-2*x)^(5/2)-375/112*(1-2*x)^(7/2)+9317/16/(1-2*x)^(1/2)+8349/8*(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {375}{112} (1-2 x)^{7/2}+\frac {335}{8} (1-2 x)^{5/2}-\frac {935}{4} (1-2 x)^{3/2}+\frac {8349}{8} \sqrt {1-2 x}+\frac {9317}{16 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

9317/(16*Sqrt[1 - 2*x]) + (8349*Sqrt[1 - 2*x])/8 - (935*(1 - 2*x)^(3/2))/4 + (335*(1 - 2*x)^(5/2))/8 - (375*(1
- 2*x)^(7/2))/112

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)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {9317}{16 (1-2 x)^{3/2}}-\frac {8349}{8 \sqrt {1-2 x}}+\frac {2805}{4} \sqrt {1-2 x}-\frac {1675}{8} (1-2 x)^{3/2}+\frac {375}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {9317}{16 \sqrt {1-2 x}}+\frac {8349}{8} \sqrt {1-2 x}-\frac {935}{4} (1-2 x)^{3/2}+\frac {335}{8} (1-2 x)^{5/2}-\frac {375}{112} (1-2 x)^{7/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} \frac {10015-9637 x-3590 x^2-1595 x^3-375 x^4}{7 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10015 - 9637*x - 3590*x^2 - 1595*x^3 - 375*x^4)/(7*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {375 x^{4}+1595 x^{3}+3590 x^{2}+9637 x -10015}{7 \sqrt {1-2 x}}\)	\(30\)
risch	\(-\frac {375 x^{4}+1595 x^{3}+3590 x^{2}+9637 x -10015}{7 \sqrt {1-2 x}}\)	\(30\)
trager	\(\frac {\left (375 x^{4}+1595 x^{3}+3590 x^{2}+9637 x -10015\right ) \sqrt {1-2 x}}{-7+14 x}\)	\(37\)
derivativedivides	\(-\frac {935 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {335 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {375 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {9317}{16 \sqrt {1-2 x}}+\frac {8349 \sqrt {1-2 x}}{8}\)	\(47\)
default	\(-\frac {935 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {335 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {375 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {9317}{16 \sqrt {1-2 x}}+\frac {8349 \sqrt {1-2 x}}{8}\)	\(47\)
meijerg	\(-\frac {54 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-351 \sqrt {\pi }+\frac {351 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {855 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-370 \sqrt {\pi }+\frac {185 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {375 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-935/4*(1-2*x)^(3/2)+335/8*(1-2*x)^(5/2)-375/112*(1-2*x)^(7/2)+9317/16/(1-2*x)^(1/2)+8349/8*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 46, normalized size = 0.70 \begin {gather*} -\frac {375}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {335}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {935}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {8349}{8} \, \sqrt {-2 \, x + 1} + \frac {9317}{16 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375/112*(-2*x + 1)^(7/2) + 335/8*(-2*x + 1)^(5/2) - 935/4*(-2*x + 1)^(3/2) + 8349/8*sqrt(-2*x + 1) + 9317/16/
sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.60, size = 36, normalized size = 0.55 \begin {gather*} \frac {{\left (375 \, x^{4} + 1595 \, x^{3} + 3590 \, x^{2} + 9637 \, x - 10015\right )} \sqrt {-2 \, x + 1}}{7 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(375*x^4 + 1595*x^3 + 3590*x^2 + 9637*x - 10015)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]
time = 11.88, size = 58, normalized size = 0.88 \begin {gather*} - \frac {375 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {335 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {935 \left (1 - 2 x\right )^{\frac {3}{2}}}{4} + \frac {8349 \sqrt {1 - 2 x}}{8} + \frac {9317}{16 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375*(1 - 2*x)**(7/2)/112 + 335*(1 - 2*x)**(5/2)/8 - 935*(1 - 2*x)**(3/2)/4 + 8349*sqrt(1 - 2*x)/8 + 9317/(16*
sqrt(1 - 2*x))

________________________________________________________________________________________

Giac [A]
time = 1.56, size = 60, normalized size = 0.91 \begin {gather*} \frac {375}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {335}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {935}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {8349}{8} \, \sqrt {-2 \, x + 1} + \frac {9317}{16 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

375/112*(2*x - 1)^3*sqrt(-2*x + 1) + 335/8*(2*x - 1)^2*sqrt(-2*x + 1) - 935/4*(-2*x + 1)^(3/2) + 8349/8*sqrt(-
2*x + 1) + 9317/16/sqrt(-2*x + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {9317}{16\,\sqrt {1-2\,x}}+\frac {8349\,\sqrt {1-2\,x}}{8}-\frac {935\,{\left (1-2\,x\right )}^{3/2}}{4}+\frac {335\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {375\,{\left (1-2\,x\right )}^{7/2}}{112} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9317/(16*(1 - 2*x)^(1/2)) + (8349*(1 - 2*x)^(1/2))/8 - (935*(1 - 2*x)^(3/2))/4 + (335*(1 - 2*x)^(5/2))/8 - (37
5*(1 - 2*x)^(7/2))/112

________________________________________________________________________________________