3.2003 \(\int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx\)

Optimal. Leaf size=66 \[ -\frac {15}{16} (1-2 x)^{9/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {3773}{16} \sqrt {1-2 x} \]

[Out]

3283/24*(1-2*x)^(3/2)-1071/20*(1-2*x)^(5/2)+621/56*(1-2*x)^(7/2)-15/16*(1-2*x)^(9/2)-3773/16*(1-2*x)^(1/2)

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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 = {77} \[ -\frac {15}{16} (1-2 x)^{9/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {3773}{16} \sqrt {1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3773*Sqrt[1 - 2*x])/16 + (3283*(1 - 2*x)^(3/2))/24 - (1071*(1 - 2*x)^(5/2))/20 + (621*(1 - 2*x)^(7/2))/56 -
(15*(1 - 2*x)^(9/2))/16

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 \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {3773}{16 \sqrt {1-2 x}}-\frac {3283}{8} \sqrt {1-2 x}+\frac {1071}{4} (1-2 x)^{3/2}-\frac {621}{8} (1-2 x)^{5/2}+\frac {135}{16} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {3773}{16} \sqrt {1-2 x}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {15}{16} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \[ -\frac {1}{105} \sqrt {1-2 x} \left (1575 x^4+6165 x^3+10881 x^2+12434 x+14954\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(Sqrt[1 - 2*x]*(14954 + 12434*x + 10881*x^2 + 6165*x^3 + 1575*x^4))

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fricas [A]  time = 1.08, size = 29, normalized size = 0.44 \[ -\frac {1}{105} \, {\left (1575 \, x^{4} + 6165 \, x^{3} + 10881 \, x^{2} + 12434 \, x + 14954\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(1575*x^4 + 6165*x^3 + 10881*x^2 + 12434*x + 14954)*sqrt(-2*x + 1)

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giac [A]  time = 1.09, size = 67, normalized size = 1.02 \[ -\frac {15}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {621}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1071}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {3283}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3773}{16} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/16*(2*x - 1)^4*sqrt(-2*x + 1) - 621/56*(2*x - 1)^3*sqrt(-2*x + 1) - 1071/20*(2*x - 1)^2*sqrt(-2*x + 1) + 3
283/24*(-2*x + 1)^(3/2) - 3773/16*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 30, normalized size = 0.45 \[ -\frac {\left (1575 x^{4}+6165 x^{3}+10881 x^{2}+12434 x +14954\right ) \sqrt {-2 x +1}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(1575*x^4+6165*x^3+10881*x^2+12434*x+14954)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.48, size = 46, normalized size = 0.70 \[ -\frac {15}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {621}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1071}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {3283}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3773}{16} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/16*(-2*x + 1)^(9/2) + 621/56*(-2*x + 1)^(7/2) - 1071/20*(-2*x + 1)^(5/2) + 3283/24*(-2*x + 1)^(3/2) - 3773
/16*sqrt(-2*x + 1)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \[ \frac {3283\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {3773\,\sqrt {1-2\,x}}{16}-\frac {1071\,{\left (1-2\,x\right )}^{5/2}}{20}+\frac {621\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {15\,{\left (1-2\,x\right )}^{9/2}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3283*(1 - 2*x)^(3/2))/24 - (3773*(1 - 2*x)^(1/2))/16 - (1071*(1 - 2*x)^(5/2))/20 + (621*(1 - 2*x)^(7/2))/56 -
(15*(1 - 2*x)^(9/2))/16

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sympy [A]  time = 44.15, size = 58, normalized size = 0.88 \[ - \frac {15 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} + \frac {621 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {1071 \left (1 - 2 x\right )^{\frac {5}{2}}}{20} + \frac {3283 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {3773 \sqrt {1 - 2 x}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15*(1 - 2*x)**(9/2)/16 + 621*(1 - 2*x)**(7/2)/56 - 1071*(1 - 2*x)**(5/2)/20 + 3283*(1 - 2*x)**(3/2)/24 - 3773
*sqrt(1 - 2*x)/16

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