3.22.49 \(\int \frac {(2+3 x)^2 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\) [2149]

Optimal. Leaf size=66 \[ \frac {5929}{48 (1-2 x)^{3/2}}-\frac {1309}{2 \sqrt {1-2 x}}-\frac {3467}{8} \sqrt {1-2 x}+\frac {85}{2} (1-2 x)^{3/2}-\frac {45}{16} (1-2 x)^{5/2} \]

[Out]

5929/48/(1-2*x)^(3/2)+85/2*(1-2*x)^(3/2)-45/16*(1-2*x)^(5/2)-1309/2/(1-2*x)^(1/2)-3467/8*(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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \begin {gather*} -\frac {45}{16} (1-2 x)^{5/2}+\frac {85}{2} (1-2 x)^{3/2}-\frac {3467}{8} \sqrt {1-2 x}-\frac {1309}{2 \sqrt {1-2 x}}+\frac {5929}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

5929/(48*(1 - 2*x)^(3/2)) - 1309/(2*Sqrt[1 - 2*x]) - (3467*Sqrt[1 - 2*x])/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1
- 2*x)^(5/2))/16

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {5929}{16 (1-2 x)^{5/2}}-\frac {1309}{2 (1-2 x)^{3/2}}+\frac {3467}{8 \sqrt {1-2 x}}-\frac {255}{2} \sqrt {1-2 x}+\frac {225}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {5929}{48 (1-2 x)^{3/2}}-\frac {1309}{2 \sqrt {1-2 x}}-\frac {3467}{8} \sqrt {1-2 x}+\frac {85}{2} (1-2 x)^{3/2}-\frac {45}{16} (1-2 x)^{5/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {2774-8430 x+3873 x^2+750 x^3+135 x^4}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2774 - 8430*x + 3873*x^2 + 750*x^3 + 135*x^4)/(1 - 2*x)^(3/2)

________________________________________________________________________________________

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

method	result	size
gosper	\(-\frac {135 x^{4}+750 x^{3}+3873 x^{2}-8430 x +2774}{3 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(30\)
trager	\(-\frac {\left (135 x^{4}+750 x^{3}+3873 x^{2}-8430 x +2774\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\)	\(37\)
risch	\(\frac {135 x^{4}+750 x^{3}+3873 x^{2}-8430 x +2774}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(37\)
derivativedivides	\(\frac {5929}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {85 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {45 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {1309}{2 \sqrt {1-2 x}}-\frac {3467 \sqrt {1-2 x}}{8}\)	\(47\)
default	\(\frac {5929}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {85 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {45 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {1309}{2 \sqrt {1-2 x}}-\frac {3467 \sqrt {1-2 x}}{8}\)	\(47\)
meijerg	\(-\frac {24 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {76 \sqrt {\pi }-\frac {19 \sqrt {\pi }\, \left (-24 x +8\right )}{2 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {541 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{6 \sqrt {\pi }}+\frac {380 \sqrt {\pi }-\frac {95 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{32 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {75 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5929/48/(1-2*x)^(3/2)+85/2*(1-2*x)^(3/2)-45/16*(1-2*x)^(5/2)-1309/2/(1-2*x)^(1/2)-3467/8*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 42, normalized size = 0.64 \begin {gather*} -\frac {45}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {85}{2} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3467}{8} \, \sqrt {-2 \, x + 1} + \frac {77 \, {\left (816 \, x - 331\right )}}{48 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/16*(-2*x + 1)^(5/2) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) + 77/48*(816*x - 331)/(-2*x + 1)^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 0.97, size = 41, normalized size = 0.62 \begin {gather*} -\frac {{\left (135 \, x^{4} + 750 \, x^{3} + 3873 \, x^{2} - 8430 \, x + 2774\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((2+3*x)^2*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(135*x^4 + 750*x^3 + 3873*x^2 - 8430*x + 2774)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

________________________________________________________________________________________

Sympy [A]
time = 11.55, size = 58, normalized size = 0.88 \begin {gather*} - \frac {45 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {85 \left (1 - 2 x\right )^{\frac {3}{2}}}{2} - \frac {3467 \sqrt {1 - 2 x}}{8} - \frac {1309}{2 \sqrt {1 - 2 x}} + \frac {5929}{48 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*(1 - 2*x)**(5/2)/16 + 85*(1 - 2*x)**(3/2)/2 - 3467*sqrt(1 - 2*x)/8 - 1309/(2*sqrt(1 - 2*x)) + 5929/(48*(1
- 2*x)**(3/2))

________________________________________________________________________________________

Giac [A]
time = 1.65, size = 56, normalized size = 0.85 \begin {gather*} -\frac {45}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {85}{2} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3467}{8} \, \sqrt {-2 \, x + 1} - \frac {77 \, {\left (816 \, x - 331\right )}}{48 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/16*(2*x - 1)^2*sqrt(-2*x + 1) + 85/2*(-2*x + 1)^(3/2) - 3467/8*sqrt(-2*x + 1) - 77/48*(816*x - 331)/((2*x
- 1)*sqrt(-2*x + 1))

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 41, normalized size = 0.62 \begin {gather*} \frac {1309\,x-\frac {25487}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {3467\,\sqrt {1-2\,x}}{8}+\frac {85\,{\left (1-2\,x\right )}^{3/2}}{2}-\frac {45\,{\left (1-2\,x\right )}^{5/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1309*x - 25487/48)/(1 - 2*x)^(3/2) - (3467*(1 - 2*x)^(1/2))/8 + (85*(1 - 2*x)^(3/2))/2 - (45*(1 - 2*x)^(5/2))
/16

________________________________________________________________________________________