3.313 \(\int x^3 (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=72 \[ \frac{6 a^2 (a+b x)^{13/2}}{13 b^4}-\frac{2 a^3 (a+b x)^{11/2}}{11 b^4}+\frac{2 (a+b x)^{17/2}}{17 b^4}-\frac{2 a (a+b x)^{15/2}}{5 b^4} \]

[Out]

(-2*a^3*(a + b*x)^(11/2))/(11*b^4) + (6*a^2*(a + b*x)^(13/2))/(13*b^4) - (2*a*(a + b*x)^(15/2))/(5*b^4) + (2*(
a + b*x)^(17/2))/(17*b^4)

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Rubi [A]  time = 0.0177268, antiderivative size = 72, 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} \[ \frac{6 a^2 (a+b x)^{13/2}}{13 b^4}-\frac{2 a^3 (a+b x)^{11/2}}{11 b^4}+\frac{2 (a+b x)^{17/2}}{17 b^4}-\frac{2 a (a+b x)^{15/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(9/2),x]

[Out]

(-2*a^3*(a + b*x)^(11/2))/(11*b^4) + (6*a^2*(a + b*x)^(13/2))/(13*b^4) - (2*a*(a + b*x)^(15/2))/(5*b^4) + (2*(
a + b*x)^(17/2))/(17*b^4)

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

Mathematica [A]  time = 0.0611463, size = 46, normalized size = 0.64 \[ \frac{2 (a+b x)^{11/2} \left (88 a^2 b x-16 a^3-286 a b^2 x^2+715 b^3 x^3\right )}{12155 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-16*a^3 + 88*a^2*b*x - 286*a*b^2*x^2 + 715*b^3*x^3))/(12155*b^4)

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Maple [A]  time = 0.005, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-1430\,{b}^{3}{x}^{3}+572\,a{b}^{2}{x}^{2}-176\,{a}^{2}bx+32\,{a}^{3}}{12155\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(9/2),x)

[Out]

-2/12155*(b*x+a)^(11/2)*(-715*b^3*x^3+286*a*b^2*x^2-88*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.00772, size = 76, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{17}{2}}}{17 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{15}{2}} a}{5 \, b^{4}} + \frac{6 \,{\left (b x + a\right )}^{\frac{13}{2}} a^{2}}{13 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{3}}{11 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/17*(b*x + a)^(17/2)/b^4 - 2/5*(b*x + a)^(15/2)*a/b^4 + 6/13*(b*x + a)^(13/2)*a^2/b^4 - 2/11*(b*x + a)^(11/2)
*a^3/b^4

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Fricas [A]  time = 1.46976, size = 227, normalized size = 3.15 \begin{align*} \frac{2 \,{\left (715 \, b^{8} x^{8} + 3289 \, a b^{7} x^{7} + 5808 \, a^{2} b^{6} x^{6} + 4714 \, a^{3} b^{5} x^{5} + 1515 \, a^{4} b^{4} x^{4} + 5 \, a^{5} b^{3} x^{3} - 6 \, a^{6} b^{2} x^{2} + 8 \, a^{7} b x - 16 \, a^{8}\right )} \sqrt{b x + a}}{12155 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/12155*(715*b^8*x^8 + 3289*a*b^7*x^7 + 5808*a^2*b^6*x^6 + 4714*a^3*b^5*x^5 + 1515*a^4*b^4*x^4 + 5*a^5*b^3*x^3
- 6*a^6*b^2*x^2 + 8*a^7*b*x - 16*a^8)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 24.4038, size = 190, normalized size = 2.64 \begin{align*} \begin{cases} - \frac{32 a^{8} \sqrt{a + b x}}{12155 b^{4}} + \frac{16 a^{7} x \sqrt{a + b x}}{12155 b^{3}} - \frac{12 a^{6} x^{2} \sqrt{a + b x}}{12155 b^{2}} + \frac{2 a^{5} x^{3} \sqrt{a + b x}}{2431 b} + \frac{606 a^{4} x^{4} \sqrt{a + b x}}{2431} + \frac{9428 a^{3} b x^{5} \sqrt{a + b x}}{12155} + \frac{1056 a^{2} b^{2} x^{6} \sqrt{a + b x}}{1105} + \frac{46 a b^{3} x^{7} \sqrt{a + b x}}{85} + \frac{2 b^{4} x^{8} \sqrt{a + b x}}{17} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(9/2),x)

[Out]

Piecewise((-32*a**8*sqrt(a + b*x)/(12155*b**4) + 16*a**7*x*sqrt(a + b*x)/(12155*b**3) - 12*a**6*x**2*sqrt(a +
b*x)/(12155*b**2) + 2*a**5*x**3*sqrt(a + b*x)/(2431*b) + 606*a**4*x**4*sqrt(a + b*x)/2431 + 9428*a**3*b*x**5*s
qrt(a + b*x)/12155 + 1056*a**2*b**2*x**6*sqrt(a + b*x)/1105 + 46*a*b**3*x**7*sqrt(a + b*x)/85 + 2*b**4*x**8*sq
rt(a + b*x)/17, Ne(b, 0)), (a**(9/2)*x**4/4, True))

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Giac [B]  time = 1.27581, size = 514, normalized size = 7.14 \begin{align*} \frac{2 \,{\left (\frac{2431 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{4}}{b^{3}} + \frac{884 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} a^{3}}{b^{3}} + \frac{510 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )} a^{2}}{b^{3}} + \frac{68 \,{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}\right )} a}{b^{3}} + \frac{7 \,{\left (6435 \,{\left (b x + a\right )}^{\frac{17}{2}} - 51051 \,{\left (b x + a\right )}^{\frac{15}{2}} a + 176715 \,{\left (b x + a\right )}^{\frac{13}{2}} a^{2} - 348075 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{3} + 425425 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{4} - 328185 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{5} + 153153 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{6} - 36465 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{7}\right )}}{b^{3}}\right )}}{765765 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/765765*(2431*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3
)*a^4/b^3 + 884*(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^3/b^3 + 510*(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^2/b^3 + 68
*(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/b^3 + 7*(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)/b^3)/b