3.4.12 \(\int x^4 (a+b x)^{9/2} \, dx\) [312]

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

[Out]

2/11*a^4*(b*x+a)^(11/2)/b^5-8/13*a^3*(b*x+a)^(13/2)/b^5+4/5*a^2*(b*x+a)^(15/2)/b^5-8/17*a*(b*x+a)^(17/2)/b^5+2
/19*(b*x+a)^(19/2)/b^5

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Rubi [A]
time = 0.02, antiderivative size = 91, 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^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 {2 (a+b x)^{19/2}}{19 b^5}-\frac {8 a (a+b x)^{17/2}}{17 b^5} \end {gather*}

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 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^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*}

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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.63 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (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\right )}{230945 b^5} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.55, size = 120, normalized size = 1.32 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (128 a^9-64 a^8 b x+48 a^7 b^2 x^2-40 a^6 b^3 x^3+35 a^5 b^4 x^4+b^5 x^5 \left (23063 a^4+75086 a^3 b x+95238 a^2 b^2 x^2+55055 a b^3 x^3+12155 b^4 x^4\right )\right ) \sqrt {a+b x}}{230945 b^5},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^5}{5}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^4*(a + b*x)^(9/2),x]')

[Out]

Piecewise[{{2 (128 a ^ 9 - 64 a ^ 8 b x + 48 a ^ 7 b ^ 2 x ^ 2 - 40 a ^ 6 b ^ 3 x ^ 3 + 35 a ^ 5 b ^ 4 x ^ 4 +
b ^ 5 x ^ 5 (23063 a ^ 4 + 75086 a ^ 3 b x + 95238 a ^ 2 b ^ 2 x ^ 2 + 55055 a b ^ 3 x ^ 3 + 12155 b ^ 4 x ^
4)) Sqrt[a + b x] / (230945 b ^ 5), b != 0}}, a ^ (9 / 2) x ^ 5 / 5]

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Maple [A]
time = 0.09, size = 62, normalized size = 0.68

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (12155 b^{4} x^{4}-5720 a \,b^{3} x^{3}+2288 a^{2} b^{2} x^{2}-704 a^{3} b x +128 a^{4}\right )}{230945 b^{5}}\)	\(54\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {8 a \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {4 a^{2} \left (b x +a \right )^{\frac {15}{2}}}{5}-\frac {8 a^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{4} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{5}}\)	\(62\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {8 a \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {4 a^{2} \left (b x +a \right )^{\frac {15}{2}}}{5}-\frac {8 a^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{4} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{5}}\)	\(62\)
trager	\(\frac {2 \left (12155 b^{9} x^{9}+55055 a \,b^{8} x^{8}+95238 a^{2} b^{7} x^{7}+75086 a^{3} b^{6} x^{6}+23063 a^{4} b^{5} x^{5}+35 a^{5} b^{4} x^{4}-40 a^{6} b^{3} x^{3}+48 a^{7} b^{2} x^{2}-64 a^{8} b x +128 a^{9}\right ) \sqrt {b x +a}}{230945 b^{5}}\)	\(109\)
risch	\(\frac {2 \left (12155 b^{9} x^{9}+55055 a \,b^{8} x^{8}+95238 a^{2} b^{7} x^{7}+75086 a^{3} b^{6} x^{6}+23063 a^{4} b^{5} x^{5}+35 a^{5} b^{4} x^{4}-40 a^{6} b^{3} x^{3}+48 a^{7} b^{2} x^{2}-64 a^{8} b x +128 a^{9}\right ) \sqrt {b x +a}}{230945 b^{5}}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/b^5*(1/19*(b*x+a)^(19/2)-4/17*a*(b*x+a)^(17/2)+2/5*a^2*(b*x+a)^(15/2)-4/13*a^3*(b*x+a)^(13/2)+1/11*a^4*(b*x+
a)^(11/2))

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Maxima [A]
time = 0.26, size = 71, normalized size = 0.78 \begin {gather*} \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 {gather*}

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 = 0.32, size = 108, normalized size = 1.19 \begin {gather*} \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 {gather*}

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 = 1.20, size = 212, normalized size = 2.33 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (71) = 142\).
time = 0.00, size = 1004, normalized size = 11.03 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{19} \sqrt {a+b x} \left (a+b x\right )^{9}-\frac {9}{17} \sqrt {a+b x} \left (a+b x\right )^{8} a+\frac {12}{5} \sqrt {a+b x} \left (a+b x\right )^{7} a^{2}-\frac {84}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{3}+\frac {126}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{4}-14 \sqrt {a+b x} \left (a+b x\right )^{4} a^{5}+12 \sqrt {a+b x} \left (a+b x\right )^{3} a^{6}-\frac {36}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{7}+3 \sqrt {a+b x} \left (a+b x\right ) a^{8}-\sqrt {a+b x} a^{9}\right )}{b^{9}}+\frac {10 a b^{4} \left (\frac {1}{17} \sqrt {a+b x} \left (a+b x\right )^{8}-\frac {8}{15} \sqrt {a+b x} \left (a+b x\right )^{7} a+\frac {28}{13} \sqrt {a+b x} \left (a+b x\right )^{6} a^{2}-\frac {56}{11} \sqrt {a+b x} \left (a+b x\right )^{5} a^{3}+\frac {70}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a^{4}-8 \sqrt {a+b x} \left (a+b x\right )^{3} a^{5}+\frac {28}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{6}-\frac {8}{3} \sqrt {a+b x} \left (a+b x\right ) a^{7}+\sqrt {a+b x} a^{8}\right )}{b^{8}}+\frac {20 a^{2} b^{3} \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 {20 a^{3} b^{2} \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^{4} b \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 {2 a^{5} \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}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(9/2),x)

[Out]

2/14549535*(46189*(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^5/b^4 + 104975*(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^4/b^4 + 48450*(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)*a^3/b^4 + 22610*(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)*a^2/b^4 + 665*(6435*(b*x + a)^(17/2
) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*(b*x + a)^(9
/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b
*x + a)*a^8)*a/b^4 + 63*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 14
92260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 3233230*(b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7
/2)*a^6 - 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8 - 230945*sqrt(b*x + a)*a^9)/b^4)/b

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Mupad [B]
time = 0.02, size = 71, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^5}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}-\frac {8\,a^3\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {4\,a^2\,{\left (a+b\,x\right )}^{15/2}}{5\,b^5}-\frac {8\,a\,{\left (a+b\,x\right )}^{17/2}}{17\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(19/2))/(19*b^5) + (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)

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