3.4.13 \(\int x^3 (a+b x)^{9/2} \, dx\) [313]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 72, 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^3 (a+b x)^{11/2}}{11 b^4}+\frac {6 a^2 (a+b x)^{13/2}}{13 b^4}+\frac {2 (a+b x)^{17/2}}{17 b^4}-\frac {2 a (a+b x)^{15/2}}{5 b^4} \end {gather*}

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 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^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*}

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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.64 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (-16 a^3+88 a^2 b x-286 a b^2 x^2+715 b^3 x^3\right )}{12155 b^4} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.33, size = 109, normalized size = 1.51 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-16 a^8+8 a^7 b x-6 a^6 b^2 x^2+5 a^5 b^3 x^3+b^4 x^4 \left (1515 a^4+4714 a^3 b x+5808 a^2 b^2 x^2+3289 a b^3 x^3+715 b^4 x^4\right )\right ) \sqrt {a+b x}}{12155 b^4},b\text {!=}0\right \}\right \},\frac {a^{\frac {9}{2}} x^4}{4}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^3*(a + b*x)^(9/2),x]')

[Out]

Piecewise[{{2 (-16 a ^ 8 + 8 a ^ 7 b x - 6 a ^ 6 b ^ 2 x ^ 2 + 5 a ^ 5 b ^ 3 x ^ 3 + b ^ 4 x ^ 4 (1515 a ^ 4 +
4714 a ^ 3 b x + 5808 a ^ 2 b ^ 2 x ^ 2 + 3289 a b ^ 3 x ^ 3 + 715 b ^ 4 x ^ 4)) Sqrt[a + b x] / (12155 b ^ 4
), b != 0}}, a ^ (9 / 2) x ^ 4 / 4]

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Maple [A]
time = 0.09, size = 50, normalized size = 0.69

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-715 b^{3} x^{3}+286 a \,b^{2} x^{2}-88 a^{2} b x +16 a^{3}\right )}{12155 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {2 a \left (b x +a \right )^{\frac {15}{2}}}{5}+\frac {6 a^{2} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{3} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {17}{2}}}{17}-\frac {2 a \left (b x +a \right )^{\frac {15}{2}}}{5}+\frac {6 a^{2} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 a^{3} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{4}}\)	\(50\)
trager	\(-\frac {2 \left (-715 b^{8} x^{8}-3289 a \,b^{7} x^{7}-5808 a^{2} x^{6} b^{6}-4714 a^{3} x^{5} b^{5}-1515 a^{4} x^{4} b^{4}-5 a^{5} x^{3} b^{3}+6 a^{6} x^{2} b^{2}-8 a^{7} x b +16 a^{8}\right ) \sqrt {b x +a}}{12155 b^{4}}\)	\(98\)
risch	\(-\frac {2 \left (-715 b^{8} x^{8}-3289 a \,b^{7} x^{7}-5808 a^{2} x^{6} b^{6}-4714 a^{3} x^{5} b^{5}-1515 a^{4} x^{4} b^{4}-5 a^{5} x^{3} b^{3}+6 a^{6} x^{2} b^{2}-8 a^{7} x b +16 a^{8}\right ) \sqrt {b x +a}}{12155 b^{4}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 56, normalized size = 0.78 \begin {gather*} \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 {gather*}

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 = 0.31, size = 97, normalized size = 1.35 \begin {gather*} \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 {gather*}

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 = 1.07, size = 190, normalized size = 2.64 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (56) = 112\).
time = 0.00, size = 866, normalized size = 12.03 \begin {gather*} \frac {\frac {2 b^{5} \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 {10 a b^{4} \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^{2} b^{3} \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^{3} b^{2} \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 {10 a^{4} b \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 {2 a^{5} \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}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(21879*(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^5
/b^3 + 12155*(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^4/b^3 + 11050*(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^3/b^3 + 2550*(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^2/b^3 + 595*(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/b^3 + 7*(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 - 875
160*(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)/
b^3)/b

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Mupad [B]
time = 0.04, size = 56, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{17/2}}{17\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4}+\frac {6\,a^2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}-\frac {2\,a\,{\left (a+b\,x\right )}^{15/2}}{5\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(17/2))/(17*b^4) - (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)

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