3.22.35 \(\int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{5/2}} \, dx\) [2135]

Optimal. Leaf size=92 \[ \frac {184877}{192 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645}{64} \sqrt {1-2 x}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {135}{64} (1-2 x)^{9/2} \]

[Out]

184877/192/(1-2*x)^(3/2)+12495/8*(1-2*x)^(3/2)-19467/64*(1-2*x)^(5/2)+1053/28*(1-2*x)^(7/2)-135/64*(1-2*x)^(9/
2)-60025/8/(1-2*x)^(1/2)-519645/64*(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 92, 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 {135}{64} (1-2 x)^{9/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {519645}{64} \sqrt {1-2 x}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {184877}{192 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

184877/(192*(1 - 2*x)^(3/2)) - 60025/(8*Sqrt[1 - 2*x]) - (519645*Sqrt[1 - 2*x])/64 + (12495*(1 - 2*x)^(3/2))/8
- (19467*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

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)^5 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {184877}{64 (1-2 x)^{5/2}}-\frac {60025}{8 (1-2 x)^{3/2}}+\frac {519645}{64 \sqrt {1-2 x}}-\frac {37485}{8} \sqrt {1-2 x}+\frac {97335}{64} (1-2 x)^{3/2}-\frac {1053}{4} (1-2 x)^{5/2}+\frac {1215}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {184877}{192 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645}{64} \sqrt {1-2 x}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {135}{64} (1-2 x)^{9/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {280696-844104 x+412812 x^2+114084 x^3+49653 x^4+16767 x^5+2835 x^6}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(280696 - 844104*x + 412812*x^2 + 114084*x^3 + 49653*x^4 + 16767*x^5 + 2835*x^6)/(1 - 2*x)^(3/2)

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696}{21 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(40\)
trager	\(-\frac {\left (2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696\right ) \sqrt {1-2 x}}{21 \left (-1+2 x \right )^{2}}\)	\(47\)
risch	\(\frac {2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696}{21 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(47\)
derivativedivides	\(\frac {184877}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {12495 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {19467 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {135 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645 \sqrt {1-2 x}}{64}\)	\(65\)
default	\(\frac {184877}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {12495 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {19467 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {135 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645 \sqrt {1-2 x}}{64}\)	\(65\)
meijerg	\(-\frac {64 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {880 \sqrt {\pi }}{3}-\frac {110 \sqrt {\pi }\, \left (-24 x +8\right )}{3 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {560 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {4560 \sqrt {\pi }-\frac {285 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {1305 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }}+\frac {\frac {12744 \sqrt {\pi }}{7}-\frac {1593 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {405 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

184877/192/(1-2*x)^(3/2)+12495/8*(1-2*x)^(3/2)-19467/64*(1-2*x)^(5/2)+1053/28*(1-2*x)^(7/2)-135/64*(1-2*x)^(9/
2)-60025/8/(1-2*x)^(1/2)-519645/64*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.32, size = 60, normalized size = 0.65 \begin {gather*} -\frac {135}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1053}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {19467}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} + \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-135/64*(-2*x + 1)^(9/2) + 1053/28*(-2*x + 1)^(7/2) - 19467/64*(-2*x + 1)^(5/2) + 12495/8*(-2*x + 1)^(3/2) - 5
19645/64*sqrt(-2*x + 1) + 2401/192*(1200*x - 523)/(-2*x + 1)^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 1.36, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (2835 \, x^{6} + 16767 \, x^{5} + 49653 \, x^{4} + 114084 \, x^{3} + 412812 \, x^{2} - 844104 \, x + 280696\right )} \sqrt {-2 \, x + 1}}{21 \, {\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(2835*x^6 + 16767*x^5 + 49653*x^4 + 114084*x^3 + 412812*x^2 - 844104*x + 280696)*sqrt(-2*x + 1)/(4*x^2 -
4*x + 1)

________________________________________________________________________________________

Sympy [A]
time = 20.41, size = 82, normalized size = 0.89 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {1053 \left (1 - 2 x\right )^{\frac {7}{2}}}{28} - \frac {19467 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {12495 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {519645 \sqrt {1 - 2 x}}{64} - \frac {60025}{8 \sqrt {1 - 2 x}} + \frac {184877}{192 \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)**5*(3+5*x)/(1-2*x)**(5/2),x)

[Out]

-135*(1 - 2*x)**(9/2)/64 + 1053*(1 - 2*x)**(7/2)/28 - 19467*(1 - 2*x)**(5/2)/64 + 12495*(1 - 2*x)**(3/2)/8 - 5
19645*sqrt(1 - 2*x)/64 - 60025/(8*sqrt(1 - 2*x)) + 184877/(192*(1 - 2*x)**(3/2))

________________________________________________________________________________________

Giac [A]
time = 2.73, size = 88, normalized size = 0.96 \begin {gather*} -\frac {135}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1053}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {19467}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\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)^5*(3+5*x)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-135/64*(2*x - 1)^4*sqrt(-2*x + 1) - 1053/28*(2*x - 1)^3*sqrt(-2*x + 1) - 19467/64*(2*x - 1)^2*sqrt(-2*x + 1)
+ 12495/8*(-2*x + 1)^(3/2) - 519645/64*sqrt(-2*x + 1) - 2401/192*(1200*x - 523)/((2*x - 1)*sqrt(-2*x + 1))

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {60025\,x}{4}-\frac {1255723}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {519645\,\sqrt {1-2\,x}}{64}+\frac {12495\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {19467\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {1053\,{\left (1-2\,x\right )}^{7/2}}{28}-\frac {135\,{\left (1-2\,x\right )}^{9/2}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((60025*x)/4 - 1255723/192)/(1 - 2*x)^(3/2) - (519645*(1 - 2*x)^(1/2))/64 + (12495*(1 - 2*x)^(3/2))/8 - (19467
*(1 - 2*x)^(5/2))/64 + (1053*(1 - 2*x)^(7/2))/28 - (135*(1 - 2*x)^(9/2))/64

________________________________________________________________________________________