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

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

[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)

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Rubi [A]  time = 0.023489, antiderivative size = 91, normalized size of antiderivative = 1., 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} \[ \frac{4 a^2 (a+b x)^{15/2}}{5 b^5}-\frac{8 a^3 (a+b x)^{13/2}}{13 b^5}+\frac{2 a^4 (a+b x)^{11/2}}{11 b^5}+\frac{2 (a+b x)^{19/2}}{19 b^5}-\frac{8 a (a+b x)^{17/2}}{17 b^5} \]

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*}

Mathematica [A]  time = 0.0790657, size = 57, normalized size = 0.63 \[ \frac{2 (a+b x)^{11/2} \left (2288 a^2 b^2 x^2-704 a^3 b x+128 a^4-5720 a b^3 x^3+12155 b^4 x^4\right )}{230945 b^5} \]

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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Maple [A]  time = 0.005, size = 54, normalized size = 0.6 \begin{align*}{\frac{24310\,{x}^{4}{b}^{4}-11440\,a{x}^{3}{b}^{3}+4576\,{a}^{2}{x}^{2}{b}^{2}-1408\,{a}^{3}xb+256\,{a}^{4}}{230945\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

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 = 1.06945, size = 96, normalized size = 1.05 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.47349, size = 265, normalized size = 2.91 \begin{align*} \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{align*}

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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Sympy [A]  time = 31.1687, size = 212, normalized size = 2.33 \begin{align*} \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{align*}

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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Giac [B]  time = 1.19301, size = 595, normalized size = 6.54 \begin{align*} \text{result too large to display} \end{align*}

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*(4199*(315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5
/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4)*a^4/b^4 + 6460*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11/2)*a + 10010*(b*
x + a)^(9/2)*a^2 - 12870*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2)*a^5)*a^3/b^4 +
1938*(3003*(b*x + a)^(15/2) - 20790*(b*x + a)^(13/2)*a + 61425*(b*x + a)^(11/2)*a^2 - 100100*(b*x + a)^(9/2)*a
^3 + 96525*(b*x + a)^(7/2)*a^4 - 54054*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6)*a^2/b^4 + 532*(6435*(b
*x + a)^(17/2) - 51051*(b*x + a)^(15/2)*a + 176715*(b*x + a)^(13/2)*a^2 - 348075*(b*x + a)^(11/2)*a^3 + 425425
*(b*x + a)^(9/2)*a^4 - 328185*(b*x + a)^(7/2)*a^5 + 153153*(b*x + a)^(5/2)*a^6 - 36465*(b*x + a)^(3/2)*a^7)*a/
b^4 + 7*(109395*(b*x + a)^(19/2) - 978120*(b*x + a)^(17/2)*a + 3879876*(b*x + a)^(15/2)*a^2 - 8953560*(b*x + a
)^(13/2)*a^3 + 13226850*(b*x + a)^(11/2)*a^4 - 12932920*(b*x + a)^(9/2)*a^5 + 8314020*(b*x + a)^(7/2)*a^6 - 33
25608*(b*x + a)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8)/b^4)/b