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

Optimal. Leaf size=105 \[ \frac {405}{128} (1-2 x)^{15/2}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {2033647}{128} \sqrt {1-2 x} \]

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Rubi [A]  time = 0.02, antiderivative size = 105, 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 {405}{128} (1-2 x)^{15/2}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {2033647}{128} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2*
x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(1 - 2*x)^(15/2))/128

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)^5 (3+5 x)^2}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {2033647}{128 \sqrt {1-2 x}}-\frac {6206585}{128} \sqrt {1-2 x}+\frac {8117095}{128} (1-2 x)^{3/2}-\frac {5896905}{128} (1-2 x)^{5/2}+\frac {2570085}{128} (1-2 x)^{7/2}-\frac {672003}{128} (1-2 x)^{9/2}+\frac {97605}{128} (1-2 x)^{11/2}-\frac {6075}{128} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {2033647}{128} \sqrt {1-2 x}+\frac {6206585}{384} (1-2 x)^{3/2}-\frac {1623419}{128} (1-2 x)^{5/2}+\frac {842415}{128} (1-2 x)^{7/2}-\frac {285565}{128} (1-2 x)^{9/2}+\frac {672003 (1-2 x)^{11/2}}{1408}-\frac {97605 (1-2 x)^{13/2}}{1664}+\frac {405}{128} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {1}{429} \sqrt {1-2 x} \left (173745 x^7+1002375 x^6+2632743 x^5+4212525 x^4+4694340 x^3+4058988 x^2+3152152 x+3275704\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/429*(Sqrt[1 - 2*x]*(3275704 + 3152152*x + 4058988*x^2 + 4694340*x^3 + 4212525*x^4 + 2632743*x^5 + 1002375*x
^6 + 173745*x^7))

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IntegrateAlgebraic [A]  time = 0.02, size = 76, normalized size = 0.72 \begin {gather*} \frac {\left (173745 (1-2 x)^7-3220965 (1-2 x)^6+26208117 (1-2 x)^5-122507385 (1-2 x)^4+361396035 (1-2 x)^3-696446751 (1-2 x)^2+887541655 (1-2 x)-872434563\right ) \sqrt {1-2 x}}{54912} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-872434563 + 887541655*(1 - 2*x) - 696446751*(1 - 2*x)^2 + 361396035*(1 - 2*x)^3 - 122507385*(1 - 2*x)^4 + 2
6208117*(1 - 2*x)^5 - 3220965*(1 - 2*x)^6 + 173745*(1 - 2*x)^7)*Sqrt[1 - 2*x])/54912

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fricas [A]  time = 1.56, size = 44, normalized size = 0.42 \begin {gather*} -\frac {1}{429} \, {\left (173745 \, x^{7} + 1002375 \, x^{6} + 2632743 \, x^{5} + 4212525 \, x^{4} + 4694340 \, x^{3} + 4058988 \, x^{2} + 3152152 \, x + 3275704\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(173745*x^7 + 1002375*x^6 + 2632743*x^5 + 4212525*x^4 + 4694340*x^3 + 4058988*x^2 + 3152152*x + 3275704
)*sqrt(-2*x + 1)

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giac [A]  time = 0.96, size = 115, normalized size = 1.10 \begin {gather*} -\frac {405}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {97605}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {672003}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {285565}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {842415}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1623419}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {6206585}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2033647}{128} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-405/128*(2*x - 1)^7*sqrt(-2*x + 1) - 97605/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 672003/1408*(2*x - 1)^5*sqrt(-2*
x + 1) - 285565/128*(2*x - 1)^4*sqrt(-2*x + 1) - 842415/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1623419/128*(2*x - 1)
^2*sqrt(-2*x + 1) + 6206585/384*(-2*x + 1)^(3/2) - 2033647/128*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 45, normalized size = 0.43 \begin {gather*} -\frac {\left (173745 x^{7}+1002375 x^{6}+2632743 x^{5}+4212525 x^{4}+4694340 x^{3}+4058988 x^{2}+3152152 x +3275704\right ) \sqrt {-2 x +1}}{429} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(173745*x^7+1002375*x^6+2632743*x^5+4212525*x^4+4694340*x^3+4058988*x^2+3152152*x+3275704)*(-2*x+1)^(1/
2)

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maxima [A]  time = 0.56, size = 73, normalized size = 0.70 \begin {gather*} \frac {405}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {97605}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {672003}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {285565}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {842415}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1623419}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {6206585}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2033647}{128} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405/128*(-2*x + 1)^(15/2) - 97605/1664*(-2*x + 1)^(13/2) + 672003/1408*(-2*x + 1)^(11/2) - 285565/128*(-2*x +
1)^(9/2) + 842415/128*(-2*x + 1)^(7/2) - 1623419/128*(-2*x + 1)^(5/2) + 6206585/384*(-2*x + 1)^(3/2) - 2033647
/128*sqrt(-2*x + 1)

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mupad [B]  time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {6206585\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {2033647\,\sqrt {1-2\,x}}{128}-\frac {1623419\,{\left (1-2\,x\right )}^{5/2}}{128}+\frac {842415\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {285565\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {672003\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {97605\,{\left (1-2\,x\right )}^{13/2}}{1664}+\frac {405\,{\left (1-2\,x\right )}^{15/2}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6206585*(1 - 2*x)^(3/2))/384 - (2033647*(1 - 2*x)^(1/2))/128 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2
*x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(1 - 2*x)^(15/2))/128

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sympy [A]  time = 98.85, size = 94, normalized size = 0.90 \begin {gather*} \frac {405 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} - \frac {97605 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} + \frac {672003 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {285565 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {842415 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} - \frac {1623419 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} + \frac {6206585 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} - \frac {2033647 \sqrt {1 - 2 x}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405*(1 - 2*x)**(15/2)/128 - 97605*(1 - 2*x)**(13/2)/1664 + 672003*(1 - 2*x)**(11/2)/1408 - 285565*(1 - 2*x)**(
9/2)/128 + 842415*(1 - 2*x)**(7/2)/128 - 1623419*(1 - 2*x)**(5/2)/128 + 6206585*(1 - 2*x)**(3/2)/384 - 2033647
*sqrt(1 - 2*x)/128

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