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

Optimal. Leaf size=92 \[ -\frac {375}{64} (1-2 x)^{9/2}+\frac {11475}{112} (1-2 x)^{7/2}-\frac {52011}{64} (1-2 x)^{5/2}+\frac {98209}{24} (1-2 x)^{3/2}-\frac {1334949}{64} \sqrt {1-2 x}-\frac {302379}{16 \sqrt {1-2 x}}+\frac {456533}{192 (1-2 x)^{3/2}} \]

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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 {375}{64} (1-2 x)^{9/2}+\frac {11475}{112} (1-2 x)^{7/2}-\frac {52011}{64} (1-2 x)^{5/2}+\frac {98209}{24} (1-2 x)^{3/2}-\frac {1334949}{64} \sqrt {1-2 x}-\frac {302379}{16 \sqrt {1-2 x}}+\frac {456533}{192 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

456533/(192*(1 - 2*x)^(3/2)) - 302379/(16*Sqrt[1 - 2*x]) - (1334949*Sqrt[1 - 2*x])/64 + (98209*(1 - 2*x)^(3/2)
)/24 - (52011*(1 - 2*x)^(5/2))/64 + (11475*(1 - 2*x)^(7/2))/112 - (375*(1 - 2*x)^(9/2))/64

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}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {456533}{64 (1-2 x)^{5/2}}-\frac {302379}{16 (1-2 x)^{3/2}}+\frac {1334949}{64 \sqrt {1-2 x}}-\frac {98209}{8} \sqrt {1-2 x}+\frac {260055}{64} (1-2 x)^{3/2}-\frac {11475}{16} (1-2 x)^{5/2}+\frac {3375}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {456533}{192 (1-2 x)^{3/2}}-\frac {302379}{16 \sqrt {1-2 x}}-\frac {1334949}{64} \sqrt {1-2 x}+\frac {98209}{24} (1-2 x)^{3/2}-\frac {52011}{64} (1-2 x)^{5/2}+\frac {11475}{112} (1-2 x)^{7/2}-\frac {375}{64} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {7875 x^6+45225 x^5+130464 x^4+293785 x^3+1051833 x^2-2146758 x+714074}{21 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(714074 - 2146758*x + 1051833*x^2 + 293785*x^3 + 130464*x^4 + 45225*x^5 + 7875*x^6)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.03, size = 67, normalized size = 0.73 \begin {gather*} \frac {-7875 (1-2 x)^6+137700 (1-2 x)^5-1092231 (1-2 x)^4+5499704 (1-2 x)^3-28033929 (1-2 x)^2-25399836 (1-2 x)+3195731}{1344 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3195731 - 25399836*(1 - 2*x) - 28033929*(1 - 2*x)^2 + 5499704*(1 - 2*x)^3 - 1092231*(1 - 2*x)^4 + 137700*(1 -
2*x)^5 - 7875*(1 - 2*x)^6)/(1344*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.33, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (7875 \, x^{6} + 45225 \, x^{5} + 130464 \, x^{4} + 293785 \, x^{3} + 1051833 \, x^{2} - 2146758 \, x + 714074\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)^3*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(7875*x^6 + 45225*x^5 + 130464*x^4 + 293785*x^3 + 1051833*x^2 - 2146758*x + 714074)*sqrt(-2*x + 1)/(4*x^
2 - 4*x + 1)

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giac [A]  time = 1.28, size = 88, normalized size = 0.96 \begin {gather*} -\frac {375}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {11475}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {52011}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {98209}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1334949}{64} \, \sqrt {-2 \, x + 1} - \frac {5929 \, {\left (1224 \, x - 535\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)^3*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-375/64*(2*x - 1)^4*sqrt(-2*x + 1) - 11475/112*(2*x - 1)^3*sqrt(-2*x + 1) - 52011/64*(2*x - 1)^2*sqrt(-2*x + 1
) + 98209/24*(-2*x + 1)^(3/2) - 1334949/64*sqrt(-2*x + 1) - 5929/192*(1224*x - 535)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \begin {gather*} -\frac {7875 x^{6}+45225 x^{5}+130464 x^{4}+293785 x^{3}+1051833 x^{2}-2146758 x +714074}{21 \left (-2 x +1\right )^{\frac {3}{2}}} \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)^(5/2),x)

[Out]

-1/21*(7875*x^6+45225*x^5+130464*x^4+293785*x^3+1051833*x^2-2146758*x+714074)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.50, size = 60, normalized size = 0.65 \begin {gather*} -\frac {375}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {11475}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {52011}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {98209}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1334949}{64} \, \sqrt {-2 \, x + 1} + \frac {5929 \, {\left (1224 \, x - 535\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)^3*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-375/64*(-2*x + 1)^(9/2) + 11475/112*(-2*x + 1)^(7/2) - 52011/64*(-2*x + 1)^(5/2) + 98209/24*(-2*x + 1)^(3/2)
- 1334949/64*sqrt(-2*x + 1) + 5929/192*(1224*x - 535)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {302379\,x}{8}-\frac {3172015}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {1334949\,\sqrt {1-2\,x}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {52011\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {11475\,{\left (1-2\,x\right )}^{7/2}}{112}-\frac {375\,{\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)^3*(5*x + 3)^3)/(1 - 2*x)^(5/2),x)

[Out]

((302379*x)/8 - 3172015/192)/(1 - 2*x)^(3/2) - (1334949*(1 - 2*x)^(1/2))/64 + (98209*(1 - 2*x)^(3/2))/24 - (52
011*(1 - 2*x)^(5/2))/64 + (11475*(1 - 2*x)^(7/2))/112 - (375*(1 - 2*x)^(9/2))/64

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sympy [A]  time = 35.49, size = 82, normalized size = 0.89 \begin {gather*} - \frac {375 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {11475 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} - \frac {52011 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {98209 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {1334949 \sqrt {1 - 2 x}}{64} - \frac {302379}{16 \sqrt {1 - 2 x}} + \frac {456533}{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)**3*(3+5*x)**3/(1-2*x)**(5/2),x)

[Out]

-375*(1 - 2*x)**(9/2)/64 + 11475*(1 - 2*x)**(7/2)/112 - 52011*(1 - 2*x)**(5/2)/64 + 98209*(1 - 2*x)**(3/2)/24
- 1334949*sqrt(1 - 2*x)/64 - 302379/(16*sqrt(1 - 2*x)) + 456533/(192*(1 - 2*x)**(3/2))

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