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

Optimal. Leaf size=92 \[ -\frac {3375}{832} (1-2 x)^{13/2}+\frac {11475}{176} (1-2 x)^{11/2}-\frac {28895}{64} (1-2 x)^{9/2}+\frac {98209}{56} (1-2 x)^{7/2}-\frac {1334949}{320} (1-2 x)^{5/2}+\frac {100793}{16} (1-2 x)^{3/2}-\frac {456533}{64} \sqrt {1-2 x} \]

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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {88} \begin {gather*} -\frac {3375}{832} (1-2 x)^{13/2}+\frac {11475}{176} (1-2 x)^{11/2}-\frac {28895}{64} (1-2 x)^{9/2}+\frac {98209}{56} (1-2 x)^{7/2}-\frac {1334949}{320} (1-2 x)^{5/2}+\frac {100793}{16} (1-2 x)^{3/2}-\frac {456533}{64} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*Sqrt[1 - 2*x])/64 + (100793*(1 - 2*x)^(3/2))/16 - (1334949*(1 - 2*x)^(5/2))/320 + (98209*(1 - 2*x)^(7
/2))/56 - (28895*(1 - 2*x)^(9/2))/64 + (11475*(1 - 2*x)^(11/2))/176 - (3375*(1 - 2*x)^(13/2))/832

Rule 88

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)^3 (3+5 x)^3}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {456533}{64 \sqrt {1-2 x}}-\frac {302379}{16} \sqrt {1-2 x}+\frac {1334949}{64} (1-2 x)^{3/2}-\frac {98209}{8} (1-2 x)^{5/2}+\frac {260055}{64} (1-2 x)^{7/2}-\frac {11475}{16} (1-2 x)^{9/2}+\frac {3375}{64} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {456533}{64} \sqrt {1-2 x}+\frac {100793}{16} (1-2 x)^{3/2}-\frac {1334949}{320} (1-2 x)^{5/2}+\frac {98209}{56} (1-2 x)^{7/2}-\frac {28895}{64} (1-2 x)^{9/2}+\frac {11475}{176} (1-2 x)^{11/2}-\frac {3375}{832} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-2 x} \left (1299375 x^6+6544125 x^5+14921900 x^4+20766885 x^3+20586249 x^2+17147586 x+18228666\right )}{5005} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5005*(Sqrt[1 - 2*x]*(18228666 + 17147586*x + 20586249*x^2 + 20766885*x^3 + 14921900*x^4 + 6544125*x^5 + 129
9375*x^6))

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IntegrateAlgebraic [A]  time = 0.02, size = 82, normalized size = 0.89 \begin {gather*} \frac {-1299375 (1-2 x)^{13/2}[phone] (1-2 x)^{11/2}-144619475 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-1336283949 (1-2 x)^{5/2}[phone] (1-2 x)^{3/2}-2284947665 \sqrt {1-2 x}}{320320} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2284947665*Sqrt[1 - 2*x] + 2017875860*(1 - 2*x)^(3/2) - 1336283949*(1 - 2*x)^(5/2) + 561755480*(1 - 2*x)^(7/
2) - 144619475*(1 - 2*x)^(9/2) + 20884500*(1 - 2*x)^(11/2) - 1299375*(1 - 2*x)^(13/2))/320320

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fricas [A]  time = 1.58, size = 39, normalized size = 0.42 \begin {gather*} -\frac {1}{5005} \, {\left (1299375 \, x^{6} + 6544125 \, x^{5} + 14921900 \, x^{4} + 20766885 \, x^{3} + 20586249 \, x^{2} + 17147586 \, x + 18228666\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(1299375*x^6 + 6544125*x^5 + 14921900*x^4 + 20766885*x^3 + 20586249*x^2 + 17147586*x + 18228666)*sqrt(
-2*x + 1)

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giac [A]  time = 1.03, size = 99, normalized size = 1.08 \begin {gather*} -\frac {3375}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {11475}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {28895}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {98209}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1334949}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {100793}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {456533}{64} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/832*(2*x - 1)^6*sqrt(-2*x + 1) - 11475/176*(2*x - 1)^5*sqrt(-2*x + 1) - 28895/64*(2*x - 1)^4*sqrt(-2*x +
1) - 98209/56*(2*x - 1)^3*sqrt(-2*x + 1) - 1334949/320*(2*x - 1)^2*sqrt(-2*x + 1) + 100793/16*(-2*x + 1)^(3/2
) - 456533/64*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {\left (1299375 x^{6}+6544125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {-2 x +1}}{5005} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5005*(1299375*x^6+6544125*x^5+14921900*x^4+20766885*x^3+20586249*x^2+17147586*x+18228666)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.46, size = 64, normalized size = 0.70 \begin {gather*} -\frac {3375}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {11475}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {28895}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {98209}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1334949}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {100793}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {456533}{64} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/832*(-2*x + 1)^(13/2) + 11475/176*(-2*x + 1)^(11/2) - 28895/64*(-2*x + 1)^(9/2) + 98209/56*(-2*x + 1)^(7
/2) - 1334949/320*(-2*x + 1)^(5/2) + 100793/16*(-2*x + 1)^(3/2) - 456533/64*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {100793\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {456533\,\sqrt {1-2\,x}}{64}-\frac {1334949\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {98209\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {28895\,{\left (1-2\,x\right )}^{9/2}}{64}+\frac {11475\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {3375\,{\left (1-2\,x\right )}^{13/2}}{832} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(100793*(1 - 2*x)^(3/2))/16 - (456533*(1 - 2*x)^(1/2))/64 - (1334949*(1 - 2*x)^(5/2))/320 + (98209*(1 - 2*x)^(
7/2))/56 - (28895*(1 - 2*x)^(9/2))/64 + (11475*(1 - 2*x)^(11/2))/176 - (3375*(1 - 2*x)^(13/2))/832

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sympy [A]  time = 78.54, size = 82, normalized size = 0.89 \begin {gather*} - \frac {3375 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {11475 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {28895 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {98209 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {1334949 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {100793 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {456533 \sqrt {1 - 2 x}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(13/2)/832 + 11475*(1 - 2*x)**(11/2)/176 - 28895*(1 - 2*x)**(9/2)/64 + 98209*(1 - 2*x)**(7/2)
/56 - 1334949*(1 - 2*x)**(5/2)/320 + 100793*(1 - 2*x)**(3/2)/16 - 456533*sqrt(1 - 2*x)/64

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