3.4.12 \(\int x^4 (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=91 \[ \frac {2 a^4 (a+b x)^{11/2}}{11 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {4 a^2 (a+b x)^{15/2}}{5 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5} \]

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Rubi [A]  time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {4 a^2 (a+b x)^{15/2}}{5 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {2 a^4 (a+b x)^{11/2}}{11 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^(9/2),x]

[Out]

(2*a^4*(a + b*x)^(11/2))/(11*b^5) - (8*a^3*(a + b*x)^(13/2))/(13*b^5) + (4*a^2*(a + b*x)^(15/2))/(5*b^5) - (8*
a*(a + b*x)^(17/2))/(17*b^5) + (2*(a + b*x)^(19/2))/(19*b^5)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int x^4 (a+b x)^{9/2} \, dx &=\int \left (\frac {a^4 (a+b x)^{9/2}}{b^4}-\frac {4 a^3 (a+b x)^{11/2}}{b^4}+\frac {6 a^2 (a+b x)^{13/2}}{b^4}-\frac {4 a (a+b x)^{15/2}}{b^4}+\frac {(a+b x)^{17/2}}{b^4}\right ) \, dx\\ &=\frac {2 a^4 (a+b x)^{11/2}}{11 b^5}-\frac {8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac {4 a^2 (a+b x)^{15/2}}{5 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5}+\frac {2 (a+b x)^{19/2}}{19 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 57, normalized size = 0.63 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (128 a^4-704 a^3 b x+2288 a^2 b^2 x^2-5720 a b^3 x^3+12155 b^4 x^4\right )}{230945 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(128*a^4 - 704*a^3*b*x + 2288*a^2*b^2*x^2 - 5720*a*b^3*x^3 + 12155*b^4*x^4))/(230945*b^5)

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IntegrateAlgebraic [A]  time = 0.03, size = 63, normalized size = 0.69 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (20995 a^4-71060 a^3 (a+b x)+92378 a^2 (a+b x)^2-54340 a (a+b x)^3+12155 (a+b x)^4\right )}{230945 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(20995*a^4 - 71060*a^3*(a + b*x) + 92378*a^2*(a + b*x)^2 - 54340*a*(a + b*x)^3 + 12155*(a
+ b*x)^4))/(230945*b^5)

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fricas [A]  time = 1.07, size = 108, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (12155 \, b^{9} x^{9} + 55055 \, a b^{8} x^{8} + 95238 \, a^{2} b^{7} x^{7} + 75086 \, a^{3} b^{6} x^{6} + 23063 \, a^{4} b^{5} x^{5} + 35 \, a^{5} b^{4} x^{4} - 40 \, a^{6} b^{3} x^{3} + 48 \, a^{7} b^{2} x^{2} - 64 \, a^{8} b x + 128 \, a^{9}\right )} \sqrt {b x + a}}{230945 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/230945*(12155*b^9*x^9 + 55055*a*b^8*x^8 + 95238*a^2*b^7*x^7 + 75086*a^3*b^6*x^6 + 23063*a^4*b^5*x^5 + 35*a^5
*b^4*x^4 - 40*a^6*b^3*x^3 + 48*a^7*b^2*x^2 - 64*a^8*b*x + 128*a^9)*sqrt(b*x + a)/b^5

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giac [B]  time = 1.12, size = 565, normalized size = 6.21 \begin {gather*} \frac {2 \, {\left (\frac {46189 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{5}}{b^{4}} + \frac {104975 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a^{4}}{b^{4}} + \frac {48450 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} a^{3}}{b^{4}} + \frac {22610 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} a^{2}}{b^{4}} + \frac {665 \, {\left (6435 \, {\left (b x + a\right )}^{\frac {17}{2}} - 58344 \, {\left (b x + a\right )}^{\frac {15}{2}} a + 235620 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{2} - 556920 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{3} + 850850 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{4} - 875160 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} + 612612 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} - 291720 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{7} + 109395 \, \sqrt {b x + a} a^{8}\right )} a}{b^{4}} + \frac {63 \, {\left (12155 \, {\left (b x + a\right )}^{\frac {19}{2}} - 122265 \, {\left (b x + a\right )}^{\frac {17}{2}} a + 554268 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{2} - 1492260 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{3} + 2645370 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{4} - 3233230 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{5} + 2771340 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} - 1662804 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{7} + 692835 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{8} - 230945 \, \sqrt {b x + a} a^{9}\right )}}{b^{4}}\right )}}{14549535 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/14549535*(46189*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*
a^3 + 315*sqrt(b*x + a)*a^4)*a^5/b^4 + 104975*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/
2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*a^4/b^4 + 48450*(231*(b*
x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)
^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*a^3/b^4 + 22610*(429*(b*x + a)^(15/2) - 3465*(
b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027
*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*a^2/b^4 + 665*(6435*(b*x + a)^(17/2
) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*(b*x + a)^(9
/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b
*x + a)*a^8)*a/b^4 + 63*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 14
92260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 3233230*(b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7
/2)*a^6 - 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8 - 230945*sqrt(b*x + a)*a^9)/b^4)/b

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maple [A]  time = 0.01, size = 54, normalized size = 0.59 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (12155 x^{4} b^{4}-5720 a \,x^{3} b^{3}+2288 a^{2} x^{2} b^{2}-704 a^{3} x b +128 a^{4}\right )}{230945 b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^(9/2),x)

[Out]

2/230945*(b*x+a)^(11/2)*(12155*b^4*x^4-5720*a*b^3*x^3+2288*a^2*b^2*x^2-704*a^3*b*x+128*a^4)/b^5

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maxima [A]  time = 1.36, size = 71, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {19}{2}}}{19 \, b^{5}} - \frac {8 \, {\left (b x + a\right )}^{\frac {17}{2}} a}{17 \, b^{5}} + \frac {4 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{2}}{5 \, b^{5}} - \frac {8 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{3}}{13 \, b^{5}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{4}}{11 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/19*(b*x + a)^(19/2)/b^5 - 8/17*(b*x + a)^(17/2)*a/b^5 + 4/5*(b*x + a)^(15/2)*a^2/b^5 - 8/13*(b*x + a)^(13/2)
*a^3/b^5 + 2/11*(b*x + a)^(11/2)*a^4/b^5

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mupad [B]  time = 0.02, size = 71, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^5}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}-\frac {8\,a^3\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {4\,a^2\,{\left (a+b\,x\right )}^{15/2}}{5\,b^5}-\frac {8\,a\,{\left (a+b\,x\right )}^{17/2}}{17\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(19/2))/(19*b^5) + (2*a^4*(a + b*x)^(11/2))/(11*b^5) - (8*a^3*(a + b*x)^(13/2))/(13*b^5) + (4*a^2
*(a + b*x)^(15/2))/(5*b^5) - (8*a*(a + b*x)^(17/2))/(17*b^5)

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sympy [A]  time = 25.70, size = 212, normalized size = 2.33 \begin {gather*} \begin {cases} \frac {256 a^{9} \sqrt {a + b x}}{230945 b^{5}} - \frac {128 a^{8} x \sqrt {a + b x}}{230945 b^{4}} + \frac {96 a^{7} x^{2} \sqrt {a + b x}}{230945 b^{3}} - \frac {16 a^{6} x^{3} \sqrt {a + b x}}{46189 b^{2}} + \frac {14 a^{5} x^{4} \sqrt {a + b x}}{46189 b} + \frac {46126 a^{4} x^{5} \sqrt {a + b x}}{230945} + \frac {13652 a^{3} b x^{6} \sqrt {a + b x}}{20995} + \frac {1332 a^{2} b^{2} x^{7} \sqrt {a + b x}}{1615} + \frac {154 a b^{3} x^{8} \sqrt {a + b x}}{323} + \frac {2 b^{4} x^{9} \sqrt {a + b x}}{19} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**(9/2),x)

[Out]

Piecewise((256*a**9*sqrt(a + b*x)/(230945*b**5) - 128*a**8*x*sqrt(a + b*x)/(230945*b**4) + 96*a**7*x**2*sqrt(a
+ b*x)/(230945*b**3) - 16*a**6*x**3*sqrt(a + b*x)/(46189*b**2) + 14*a**5*x**4*sqrt(a + b*x)/(46189*b) + 46126
*a**4*x**5*sqrt(a + b*x)/230945 + 13652*a**3*b*x**6*sqrt(a + b*x)/20995 + 1332*a**2*b**2*x**7*sqrt(a + b*x)/16
15 + 154*a*b**3*x**8*sqrt(a + b*x)/323 + 2*b**4*x**9*sqrt(a + b*x)/19, Ne(b, 0)), (a**(9/2)*x**5/5, True))

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