3.3.100 \(\int x^3 (a+b x)^{5/2} \, dx\) [300]

Optimal. Leaf size=72 \[ -\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4} \]

[Out]

-2/7*a^3*(b*x+a)^(7/2)/b^4+2/3*a^2*(b*x+a)^(9/2)/b^4-6/11*a*(b*x+a)^(11/2)/b^4+2/13*(b*x+a)^(13/2)/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)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(5/2),x]

[Out]

(-2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a + b*x)^(11/2))/(11*b^4) + (2*(a +
b*x)^(13/2))/(13*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)^{5/2} \, dx &=\int \left (-\frac {a^3 (a+b x)^{5/2}}{b^3}+\frac {3 a^2 (a+b x)^{7/2}}{b^3}-\frac {3 a (a+b x)^{9/2}}{b^3}+\frac {(a+b x)^{11/2}}{b^3}\right ) \, dx\\ &=-\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.64 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-16 a^3+56 a^2 b x-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2)*(-16*a^3 + 56*a^2*b*x - 126*a*b^2*x^2 + 231*b^3*x^3))/(3003*b^4)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.63, size = 88, normalized size = 1.22 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-16 a^6+8 a^5 b x-6 a^4 b^2 x^2+5 a^3 b^3 x^3+7 b^4 x^4 \left (53 a^2+81 a b x+33 b^2 x^2\right )\right ) \sqrt {a+b x}}{3003 b^4},b\text {!=}0\right \}\right \},\frac {a^{\frac {5}{2}} x^4}{4}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^3*(a + b*x)^(5/2),x]')

[Out]

Piecewise[{{2 (-16 a ^ 6 + 8 a ^ 5 b x - 6 a ^ 4 b ^ 2 x ^ 2 + 5 a ^ 3 b ^ 3 x ^ 3 + 7 b ^ 4 x ^ 4 (53 a ^ 2 +
81 a b x + 33 b ^ 2 x ^ 2)) Sqrt[a + b x] / (3003 b ^ 4), b != 0}}, a ^ (5 / 2) x ^ 4 / 4]

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

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{3}+126 a \,b^{2} x^{2}-56 a^{2} b x +16 a^{3}\right )}{3003 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {6 a \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 a^{2} \left (b x +a \right )^{\frac {9}{2}}}{3}-\frac {2 a^{3} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {6 a \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 a^{2} \left (b x +a \right )^{\frac {9}{2}}}{3}-\frac {2 a^{3} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{4}}\)	\(50\)
trager	\(-\frac {2 \left (-231 x^{6} b^{6}-567 a \,x^{5} b^{5}-371 a^{2} x^{4} b^{4}-5 a^{3} b^{3} x^{3}+6 a^{4} x^{2} b^{2}-8 a^{5} x b +16 a^{6}\right ) \sqrt {b x +a}}{3003 b^{4}}\)	\(76\)
risch	\(-\frac {2 \left (-231 x^{6} b^{6}-567 a \,x^{5} b^{5}-371 a^{2} x^{4} b^{4}-5 a^{3} b^{3} x^{3}+6 a^{4} x^{2} b^{2}-8 a^{5} x b +16 a^{6}\right ) \sqrt {b x +a}}{3003 b^{4}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/b^4*(1/13*(b*x+a)^(13/2)-3/11*a*(b*x+a)^(11/2)+1/3*a^2*(b*x+a)^(9/2)-1/7*a^3*(b*x+a)^(7/2))

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Maxima [A]
time = 0.25, size = 56, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{4}} - \frac {6 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{4}} + \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

2/13*(b*x + a)^(13/2)/b^4 - 6/11*(b*x + a)^(11/2)*a/b^4 + 2/3*(b*x + a)^(9/2)*a^2/b^4 - 2/7*(b*x + a)^(7/2)*a^
3/b^4

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Fricas [A]
time = 0.29, size = 75, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt {b x + a}}{3003 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^6*x^6 + 567*a*b^5*x^5 + 371*a^2*b^4*x^4 + 5*a^3*b^3*x^3 - 6*a^4*b^2*x^2 + 8*a^5*b*x - 16*a^6)*sq
rt(b*x + a)/b^4

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Sympy [A]
time = 0.41, size = 146, normalized size = 2.03 \begin {gather*} \begin {cases} - \frac {32 a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 a b x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{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)**(5/2),x)

[Out]

Piecewise((-32*a**6*sqrt(a + b*x)/(3003*b**4) + 16*a**5*x*sqrt(a + b*x)/(3003*b**3) - 4*a**4*x**2*sqrt(a + b*x
)/(1001*b**2) + 10*a**3*x**3*sqrt(a + b*x)/(3003*b) + 106*a**2*x**4*sqrt(a + b*x)/429 + 54*a*b*x**5*sqrt(a + b
*x)/143 + 2*b**2*x**6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(5/2)*x**4/4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (56) = 112\).
time = 0.00, size = 481, normalized size = 6.68 \begin {gather*} \frac {\frac {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 {6 a 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 {6 a^{2} 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^{3} \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)^(5/2),x)

[Out]

2/15015*(429*(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^
3 + 143*(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^3 + 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 + 5*(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^3)/b

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(5/2),x)

[Out]

(2*(a + b*x)^(13/2))/(13*b^4) - (2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a +
b*x)^(11/2))/(11*b^4)

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