3.4.15 \(\int x (a+b x)^{9/2} \, dx\) [315]

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

[Out]

-2/11*a*(b*x+a)^(11/2)/b^2+2/13*(b*x+a)^(13/2)/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 = {45} \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 45

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} (-2 a+11 b x)}{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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.94, size = 86, normalized size = 2.53 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-2 a^6+a^5 b x+b^2 x^2 \left (35 a^4+90 a^3 b x+100 a^2 b^2 x^2+53 a b^3 x^3+11 b^4 x^4\right )\right ) \sqrt {a+b x}}{143 b^2},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^2}{2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^1*(a + b*x)^(9/2),x]')

[Out]

Piecewise[{{2 (-2 a ^ 6 + a ^ 5 b x + b ^ 2 x ^ 2 (35 a ^ 4 + 90 a ^ 3 b x + 100 a ^ 2 b ^ 2 x ^ 2 + 53 a b ^
3 x ^ 3 + 11 b ^ 4 x ^ 4)) Sqrt[a + b x] / (143 b ^ 2), b != 0}}, a ^ (9 / 2) x ^ 2 / 2]

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Maple [A]
time = 0.10, size = 26, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-11 b x +2 a \right )}{143 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{2}}\)	\(26\)
trager	\(-\frac {2 \left (-11 x^{6} b^{6}-53 a \,x^{5} b^{5}-100 a^{2} x^{4} b^{4}-90 a^{3} b^{3} x^{3}-35 a^{4} x^{2} b^{2}-a^{5} x b +2 a^{6}\right ) \sqrt {b x +a}}{143 b^{2}}\)	\(76\)
risch	\(-\frac {2 \left (-11 x^{6} b^{6}-53 a \,x^{5} b^{5}-100 a^{2} x^{4} b^{4}-90 a^{3} b^{3} x^{3}-35 a^{4} x^{2} b^{2}-a^{5} x b +2 a^{6}\right ) \sqrt {b x +a}}{143 b^{2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^2*(1/13*(b*x+a)^(13/2)-1/11*a*(b*x+a)^(11/2))

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Maxima [A]
time = 0.27, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
time = 0.30, 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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Sympy [A]
time = 0.82, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (26) = 52\).
time = 0.00, size = 586, normalized size = 17.24 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{13} \sqrt {a+b x} \left (a+b x\right )^{6}-\frac {6}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right )^{4} a^{2}-\frac {20}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{3}+3 \sqrt {a+b x} \left (a+b x\right )^{2} a^{4}-2 \sqrt {a+b x} \left (a+b x\right ) a^{5}+\sqrt {a+b x} a^{6}\right )}{b^{6}}+\frac {10 a b^{4} \left (\frac {1}{11} \sqrt {a+b x} \left (a+b x\right )^{5}-\frac {5}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a+\frac {10}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{2}-2 \sqrt {a+b x} \left (a+b x\right )^{2} a^{3}+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right ) a^{4}-\sqrt {a+b x} a^{5}\right )}{b^{5}}+\frac {20 a^{2} b^{3} \left (\frac {1}{9} \sqrt {a+b x} \left (a+b x\right )^{4}-\frac {4}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a+\frac {6}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{2}-\frac {4}{3} \sqrt {a+b x} \left (a+b x\right ) a^{3}+\sqrt {a+b x} a^{4}\right )}{b^{4}}+\frac {20 a^{3} b^{2} \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}+\frac {10 a^{4} b \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}+\frac {2 a^{5} \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )}{b}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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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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