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

Optimal. Leaf size=79 \[ \frac {675}{352} (1-2 x)^{11/2}-\frac {855}{32} (1-2 x)^{9/2}+\frac {17541}{112} (1-2 x)^{7/2}-\frac {39977}{80} (1-2 x)^{5/2}+\frac {91091}{96} (1-2 x)^{3/2}-\frac {41503}{32} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 79, 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 {675}{352} (1-2 x)^{11/2}-\frac {855}{32} (1-2 x)^{9/2}+\frac {17541}{112} (1-2 x)^{7/2}-\frac {39977}{80} (1-2 x)^{5/2}+\frac {91091}{96} (1-2 x)^{3/2}-\frac {41503}{32} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*Sqrt[1 - 2*x])/32 + (91091*(1 - 2*x)^(3/2))/96 - (39977*(1 - 2*x)^(5/2))/80 + (17541*(1 - 2*x)^(7/2))/
112 - (855*(1 - 2*x)^(9/2))/32 + (675*(1 - 2*x)^(11/2))/352

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)^2}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {41503}{32 \sqrt {1-2 x}}-\frac {91091}{32} \sqrt {1-2 x}+\frac {39977}{16} (1-2 x)^{3/2}-\frac {17541}{16} (1-2 x)^{5/2}+\frac {7695}{32} (1-2 x)^{7/2}-\frac {675}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {41503}{32} \sqrt {1-2 x}+\frac {91091}{96} (1-2 x)^{3/2}-\frac {39977}{80} (1-2 x)^{5/2}+\frac {17541}{112} (1-2 x)^{7/2}-\frac {855}{32} (1-2 x)^{9/2}+\frac {675}{352} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {1-2 x} \left (70875 x^5+316575 x^4+636795 x^3+790023 x^2+743822 x+826982\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1155*(Sqrt[1 - 2*x]*(826982 + 743822*x + 790023*x^2 + 636795*x^3 + 316575*x^4 + 70875*x^5))

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IntegrateAlgebraic [A]  time = 0.02, size = 71, normalized size = 0.90 \begin {gather*} \frac {70875 (1-2 x)^{11/2}-987525 (1-2 x)^{9/2}+5788530 (1-2 x)^{7/2}-18469374 (1-2 x)^{5/2}[phone] (1-2 x)^{3/2}-47935965 \sqrt {1-2 x}}{36960} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-47935965*Sqrt[1 - 2*x] + 35070035*(1 - 2*x)^(3/2) - 18469374*(1 - 2*x)^(5/2) + 5788530*(1 - 2*x)^(7/2) - 987
525*(1 - 2*x)^(9/2) + 70875*(1 - 2*x)^(11/2))/36960

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fricas [A]  time = 1.79, size = 34, normalized size = 0.43 \begin {gather*} -\frac {1}{1155} \, {\left (70875 \, x^{5} + 316575 \, x^{4} + 636795 \, x^{3} + 790023 \, x^{2} + 743822 \, x + 826982\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)^2/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1155*(70875*x^5 + 316575*x^4 + 636795*x^3 + 790023*x^2 + 743822*x + 826982)*sqrt(-2*x + 1)

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giac [A]  time = 0.92, size = 83, normalized size = 1.05 \begin {gather*} -\frac {675}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {855}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {17541}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {39977}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {91091}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {41503}{32} \, \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)^2/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-675/352*(2*x - 1)^5*sqrt(-2*x + 1) - 855/32*(2*x - 1)^4*sqrt(-2*x + 1) - 17541/112*(2*x - 1)^3*sqrt(-2*x + 1)
- 39977/80*(2*x - 1)^2*sqrt(-2*x + 1) + 91091/96*(-2*x + 1)^(3/2) - 41503/32*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (70875 x^{5}+316575 x^{4}+636795 x^{3}+790023 x^{2}+743822 x +826982\right ) \sqrt {-2 x +1}}{1155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(70875*x^5+316575*x^4+636795*x^3+790023*x^2+743822*x+826982)*(-2*x+1)^(1/2)

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maxima [A]  time = 0.46, size = 55, normalized size = 0.70 \begin {gather*} \frac {675}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {855}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {17541}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {39977}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {91091}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {41503}{32} \, \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)^2/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

675/352*(-2*x + 1)^(11/2) - 855/32*(-2*x + 1)^(9/2) + 17541/112*(-2*x + 1)^(7/2) - 39977/80*(-2*x + 1)^(5/2) +
91091/96*(-2*x + 1)^(3/2) - 41503/32*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {91091\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {41503\,\sqrt {1-2\,x}}{32}-\frac {39977\,{\left (1-2\,x\right )}^{5/2}}{80}+\frac {17541\,{\left (1-2\,x\right )}^{7/2}}{112}-\frac {855\,{\left (1-2\,x\right )}^{9/2}}{32}+\frac {675\,{\left (1-2\,x\right )}^{11/2}}{352} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(91091*(1 - 2*x)^(3/2))/96 - (41503*(1 - 2*x)^(1/2))/32 - (39977*(1 - 2*x)^(5/2))/80 + (17541*(1 - 2*x)^(7/2))
/112 - (855*(1 - 2*x)^(9/2))/32 + (675*(1 - 2*x)^(11/2))/352

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sympy [A]  time = 72.28, size = 70, normalized size = 0.89 \begin {gather*} \frac {675 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {855 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} + \frac {17541 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} - \frac {39977 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} + \frac {91091 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} - \frac {41503 \sqrt {1 - 2 x}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675*(1 - 2*x)**(11/2)/352 - 855*(1 - 2*x)**(9/2)/32 + 17541*(1 - 2*x)**(7/2)/112 - 39977*(1 - 2*x)**(5/2)/80 +
91091*(1 - 2*x)**(3/2)/96 - 41503*sqrt(1 - 2*x)/32

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