3.1858 \(\int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=92 \[ -\frac {1215 (1-2 x)^{17/2}}{1088}+\frac {351}{20} (1-2 x)^{15/2}-\frac {97335}{832} (1-2 x)^{13/2}+\frac {37485}{88} (1-2 x)^{11/2}-\frac {173215}{192} (1-2 x)^{9/2}+\frac {8575}{8} (1-2 x)^{7/2}-\frac {184877}{320} (1-2 x)^{5/2} \]

[Out]

-184877/320*(1-2*x)^(5/2)+8575/8*(1-2*x)^(7/2)-173215/192*(1-2*x)^(9/2)+37485/88*(1-2*x)^(11/2)-97335/832*(1-2
*x)^(13/2)+351/20*(1-2*x)^(15/2)-1215/1088*(1-2*x)^(17/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, 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 = {77} \[ -\frac {1215 (1-2 x)^{17/2}}{1088}+\frac {351}{20} (1-2 x)^{15/2}-\frac {97335}{832} (1-2 x)^{13/2}+\frac {37485}{88} (1-2 x)^{11/2}-\frac {173215}{192} (1-2 x)^{9/2}+\frac {8575}{8} (1-2 x)^{7/2}-\frac {184877}{320} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-184877*(1 - 2*x)^(5/2))/320 + (8575*(1 - 2*x)^(7/2))/8 - (173215*(1 - 2*x)^(9/2))/192 + (37485*(1 - 2*x)^(11
/2))/88 - (97335*(1 - 2*x)^(13/2))/832 + (351*(1 - 2*x)^(15/2))/20 - (1215*(1 - 2*x)^(17/2))/1088

Rule 77

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 (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx &=\int \left (\frac {184877}{64} (1-2 x)^{3/2}-\frac {60025}{8} (1-2 x)^{5/2}+\frac {519645}{64} (1-2 x)^{7/2}-\frac {37485}{8} (1-2 x)^{9/2}+\frac {97335}{64} (1-2 x)^{11/2}-\frac {1053}{4} (1-2 x)^{13/2}+\frac {1215}{64} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {184877}{320} (1-2 x)^{5/2}+\frac {8575}{8} (1-2 x)^{7/2}-\frac {173215}{192} (1-2 x)^{9/2}+\frac {37485}{88} (1-2 x)^{11/2}-\frac {97335}{832} (1-2 x)^{13/2}+\frac {351}{20} (1-2 x)^{15/2}-\frac {1215 (1-2 x)^{17/2}}{1088}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 43, normalized size = 0.47 \[ -\frac {(1-2 x)^{5/2} \left (2606175 x^6+12660219 x^5+26832465 x^4+32431860 x^3+24424220 x^2+11562520 x+3012632\right )}{36465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36465*((1 - 2*x)^(5/2)*(3012632 + 11562520*x + 24424220*x^2 + 32431860*x^3 + 26832465*x^4 + 12660219*x^5 +
2606175*x^6))

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 49, normalized size = 0.53 \[ -\frac {1}{36465} \, {\left (10424700 \, x^{8} + 40216176 \, x^{7} + 59295159 \, x^{6} + 35057799 \, x^{5} - 5198095 \, x^{4} - 19014940 \, x^{3} - 9775332 \, x^{2} - 488008 \, x + 3012632\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(10424700*x^8 + 40216176*x^7 + 59295159*x^6 + 35057799*x^5 - 5198095*x^4 - 19014940*x^3 - 9775332*x^2
- 488008*x + 3012632)*sqrt(-2*x + 1)

________________________________________________________________________________________

giac [A]  time = 1.43, size = 113, normalized size = 1.23 \[ -\frac {1215}{1088} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {351}{20} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {97335}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {37485}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {173215}{192} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8575}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {184877}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 351/20*(2*x - 1)^7*sqrt(-2*x + 1) - 97335/832*(2*x - 1)^6*sqrt(-2*x +
1) - 37485/88*(2*x - 1)^5*sqrt(-2*x + 1) - 173215/192*(2*x - 1)^4*sqrt(-2*x + 1) - 8575/8*(2*x - 1)^3*sqrt(-2*
x + 1) - 184877/320*(2*x - 1)^2*sqrt(-2*x + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 40, normalized size = 0.43 \[ -\frac {\left (2606175 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +3012632\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{36465} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(2606175*x^6+12660219*x^5+26832465*x^4+32431860*x^3+24424220*x^2+11562520*x+3012632)*(-2*x+1)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 64, normalized size = 0.70 \[ -\frac {1215}{1088} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {351}{20} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {97335}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {37485}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {173215}{192} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8575}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {184877}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215/1088*(-2*x + 1)^(17/2) + 351/20*(-2*x + 1)^(15/2) - 97335/832*(-2*x + 1)^(13/2) + 37485/88*(-2*x + 1)^(1
1/2) - 173215/192*(-2*x + 1)^(9/2) + 8575/8*(-2*x + 1)^(7/2) - 184877/320*(-2*x + 1)^(5/2)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \[ \frac {8575\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {184877\,{\left (1-2\,x\right )}^{5/2}}{320}-\frac {173215\,{\left (1-2\,x\right )}^{9/2}}{192}+\frac {37485\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {97335\,{\left (1-2\,x\right )}^{13/2}}{832}+\frac {351\,{\left (1-2\,x\right )}^{15/2}}{20}-\frac {1215\,{\left (1-2\,x\right )}^{17/2}}{1088} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8575*(1 - 2*x)^(7/2))/8 - (184877*(1 - 2*x)^(5/2))/320 - (173215*(1 - 2*x)^(9/2))/192 + (37485*(1 - 2*x)^(11/
2))/88 - (97335*(1 - 2*x)^(13/2))/832 + (351*(1 - 2*x)^(15/2))/20 - (1215*(1 - 2*x)^(17/2))/1088

________________________________________________________________________________________

sympy [A]  time = 21.19, size = 82, normalized size = 0.89 \[ - \frac {1215 \left (1 - 2 x\right )^{\frac {17}{2}}}{1088} + \frac {351 \left (1 - 2 x\right )^{\frac {15}{2}}}{20} - \frac {97335 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {37485 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {173215 \left (1 - 2 x\right )^{\frac {9}{2}}}{192} + \frac {8575 \left (1 - 2 x\right )^{\frac {7}{2}}}{8} - \frac {184877 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1215*(1 - 2*x)**(17/2)/1088 + 351*(1 - 2*x)**(15/2)/20 - 97335*(1 - 2*x)**(13/2)/832 + 37485*(1 - 2*x)**(11/2
)/88 - 173215*(1 - 2*x)**(9/2)/192 + 8575*(1 - 2*x)**(7/2)/8 - 184877*(1 - 2*x)**(5/2)/320

________________________________________________________________________________________