3.4.14 \(\int x^2 (a+b x)^{9/2} \, dx\) [314]

Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, 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 = {45} \begin {gather*} \frac {2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(9/2),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (8 a^2-44 a b x+143 b^2 x^2\right )}{2145 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(8*a^2 - 44*a*b*x + 143*b^2*x^2))/(2145*b^3)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.14, size = 98, normalized size = 1.85 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (8 a^7-4 a^6 b x+3 a^5 b^2 x^2+b^3 x^3 \left (355 a^4+1030 a^3 b x+1218 a^2 b^2 x^2+671 a b^3 x^3+143 b^4 x^4\right )\right ) \sqrt {a+b x}}{2145 b^3},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^3}{3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{2 (8 a ^ 7 - 4 a ^ 6 b x + 3 a ^ 5 b ^ 2 x ^ 2 + b ^ 3 x ^ 3 (355 a ^ 4 + 1030 a ^ 3 b x + 1218 a
^ 2 b ^ 2 x ^ 2 + 671 a b ^ 3 x ^ 3 + 143 b ^ 4 x ^ 4)) Sqrt[a + b x] / (2145 b ^ 3), b != 0}}, a ^ (9 / 2) x
^ 3 / 3]

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Maple [A]
time = 0.10, size = 38, normalized size = 0.72

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (143 x^{2} b^{2}-44 a b x +8 a^{2}\right )}{2145 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {15}{2}}}{15}-\frac {4 a \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {15}{2}}}{15}-\frac {4 a \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{3}}\)	\(38\)
trager	\(\frac {2 \left (143 b^{7} x^{7}+671 a \,b^{6} x^{6}+1218 a^{2} b^{5} x^{5}+1030 a^{3} b^{4} x^{4}+355 a^{4} b^{3} x^{3}+3 a^{5} b^{2} x^{2}-4 a^{6} b x +8 a^{7}\right ) \sqrt {b x +a}}{2145 b^{3}}\)	\(87\)
risch	\(\frac {2 \left (143 b^{7} x^{7}+671 a \,b^{6} x^{6}+1218 a^{2} b^{5} x^{5}+1030 a^{3} b^{4} x^{4}+355 a^{4} b^{3} x^{3}+3 a^{5} b^{2} x^{2}-4 a^{6} b x +8 a^{7}\right ) \sqrt {b x +a}}{2145 b^{3}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^3*(1/15*(b*x+a)^(15/2)-2/13*a*(b*x+a)^(13/2)+1/11*a^2*(b*x+a)^(11/2))

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Maxima [A]
time = 0.26, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}}}{15 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {13}{2}} a}{13 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2}}{11 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (41) = 82\).
time = 0.30, size = 86, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (143 \, b^{7} x^{7} + 671 \, a b^{6} x^{6} + 1218 \, a^{2} b^{5} x^{5} + 1030 \, a^{3} b^{4} x^{4} + 355 \, a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} - 4 \, a^{6} b x + 8 \, a^{7}\right )} \sqrt {b x + a}}{2145 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*(143*b^7*x^7 + 671*a*b^6*x^6 + 1218*a^2*b^5*x^5 + 1030*a^3*b^4*x^4 + 355*a^4*b^3*x^3 + 3*a^5*b^2*x^2 -
4*a^6*b*x + 8*a^7)*sqrt(b*x + a)/b^3

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Sympy [A]
time = 0.95, size = 168, normalized size = 3.17 \begin {gather*} \begin {cases} \frac {16 a^{7} \sqrt {a + b x}}{2145 b^{3}} - \frac {8 a^{6} x \sqrt {a + b x}}{2145 b^{2}} + \frac {2 a^{5} x^{2} \sqrt {a + b x}}{715 b} + \frac {142 a^{4} x^{3} \sqrt {a + b x}}{429} + \frac {412 a^{3} b x^{4} \sqrt {a + b x}}{429} + \frac {812 a^{2} b^{2} x^{5} \sqrt {a + b x}}{715} + \frac {122 a b^{3} x^{6} \sqrt {a + b x}}{195} + \frac {2 b^{4} x^{7} \sqrt {a + b x}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*a**7*sqrt(a + b*x)/(2145*b**3) - 8*a**6*x*sqrt(a + b*x)/(2145*b**2) + 2*a**5*x**2*sqrt(a + b*x)/
(715*b) + 142*a**4*x**3*sqrt(a + b*x)/429 + 412*a**3*b*x**4*sqrt(a + b*x)/429 + 812*a**2*b**2*x**5*sqrt(a + b*
x)/715 + 122*a*b**3*x**6*sqrt(a + b*x)/195 + 2*b**4*x**7*sqrt(a + b*x)/15, Ne(b, 0)), (a**(9/2)*x**3/3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (41) = 82\).
time = 0.00, size = 727, normalized size = 13.72 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{15} \sqrt {a+b x} \left (a+b x\right )^{7}-\frac {7}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a+\frac {21}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{2}-\frac {35}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{3}+5 \sqrt {a+b x} \left (a+b x\right )^{3} a^{4}-\frac {21}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{5}+\frac {7}{3} \sqrt {a+b x} \left (a+b x\right ) a^{6}-\sqrt {a+b x} a^{7}\right )}{b^{7}}+\frac {10 a b^{4} \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 {20 a^{2} b^{3} \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^{3} b^{2} \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 {10 a^{4} b \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 {2 a^{5} \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}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*a^5/b^2 + 6435*(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^4/b^2 + 1430*(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^3/b^2
+ 650*(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^2/b^2 + 75*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5
005*(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 + 300
3*sqrt(b*x + a)*a^6)*a/b^2 + 7*(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)/b^2)/b

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Mupad [B]
time = 0.04, size = 36, normalized size = 0.68 \begin {gather*} \frac {\frac {2\,{\left (a+b\,x\right )}^{15/2}}{15}-\frac {4\,a\,{\left (a+b\,x\right )}^{13/2}}{13}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{11/2}}{11}}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(9/2),x)

[Out]

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

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