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

Optimal. Leaf size=34 \[ \frac {2 (a+b x)^{13/2}}{13 b^2}-\frac {2 a (a+b x)^{11/2}}{11 b^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {2 (a+b x)^{13/2}}{13 b^2}-\frac {2 a (a+b x)^{11/2}}{11 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(9/2),x]

[Out]

(-2*a*(a + b*x)^(11/2))/(11*b^2) + (2*(a + b*x)^(13/2))/(13*b^2)

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

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Mathematica [A]  time = 0.02, size = 24, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{11/2} (11 b x-2 a)}{143 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-2*a + 11*b*x))/(143*b^2)

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IntegrateAlgebraic [B]  time = 0.01, size = 79, normalized size = 2.32 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (2 a^6-a^5 b x-35 a^4 b^2 x^2-90 a^3 b^3 x^3-100 a^2 b^4 x^4-53 a b^5 x^5-11 b^6 x^6\right )}{143 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(2*a^6 - a^5*b*x - 35*a^4*b^2*x^2 - 90*a^3*b^3*x^3 - 100*a^2*b^4*x^4 - 53*a*b^5*x^5 - 11*b^6
*x^6))/(143*b^2)

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fricas [B]  time = 0.72, size = 74, normalized size = 2.18 \begin {gather*} \frac {2 \, {\left (11 \, b^{6} x^{6} + 53 \, a b^{5} x^{5} + 100 \, a^{2} b^{4} x^{4} + 90 \, a^{3} b^{3} x^{3} + 35 \, a^{4} b^{2} x^{2} + a^{5} b x - 2 \, a^{6}\right )} \sqrt {b x + a}}{143 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/143*(11*b^6*x^6 + 53*a*b^5*x^5 + 100*a^2*b^4*x^4 + 90*a^3*b^3*x^3 + 35*a^4*b^2*x^2 + a^5*b*x - 2*a^6)*sqrt(b
*x + a)/b^2

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giac [B]  time = 1.27, size = 347, normalized size = 10.21 \begin {gather*} \frac {2 \, {\left (\frac {3003 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{5}}{b} + \frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{4}}{b} + \frac {2574 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{3}}{b} + \frac {286 \, {\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^{2}}{b} + \frac {65 \, {\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}{b} + \frac {3 \, {\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 )}}{b}\right )}}{9009 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(3003*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*a^5/b + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15
*sqrt(b*x + a)*a^2)*a^4/b + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(
b*x + a)*a^3)*a^3/b + 286*(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^2/b + 65*(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/b + 3*(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)/b)/b

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maple [A]  time = 0.00, size = 21, normalized size = 0.62 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-11 b x +2 a \right )}{143 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/143*(b*x+a)^(11/2)*(-11*b*x+2*a)/b^2

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maxima [A]  time = 1.36, size = 26, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*(b*x + a)^(13/2)/b^2 - 2/11*(b*x + a)^(11/2)*a/b^2

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mupad [B]  time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {26\,a\,{\left (a+b\,x\right )}^{11/2}-22\,{\left (a+b\,x\right )}^{13/2}}{143\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^(9/2),x)

[Out]

-(26*a*(a + b*x)^(11/2) - 22*(a + b*x)^(13/2))/(143*b^2)

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sympy [A]  time = 14.85, size = 146, normalized size = 4.29 \begin {gather*} \begin {cases} - \frac {4 a^{6} \sqrt {a + b x}}{143 b^{2}} + \frac {2 a^{5} x \sqrt {a + b x}}{143 b} + \frac {70 a^{4} x^{2} \sqrt {a + b x}}{143} + \frac {180 a^{3} b x^{3} \sqrt {a + b x}}{143} + \frac {200 a^{2} b^{2} x^{4} \sqrt {a + b x}}{143} + \frac {106 a b^{3} x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{4} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*a**6*sqrt(a + b*x)/(143*b**2) + 2*a**5*x*sqrt(a + b*x)/(143*b) + 70*a**4*x**2*sqrt(a + b*x)/143
+ 180*a**3*b*x**3*sqrt(a + b*x)/143 + 200*a**2*b**2*x**4*sqrt(a + b*x)/143 + 106*a*b**3*x**5*sqrt(a + b*x)/143
+ 2*b**4*x**6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(9/2)*x**2/2, True))

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