3.15.45 \(\int (a+b x)^5 \sqrt {a c+b c x} \, dx\) [1445]

Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{13/2}}{13 b c^6} \]

[Out]

2/13*(b*c*x+a*c)^(13/2)/b/c^6

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \begin {gather*} \frac {2 (a c+b c x)^{13/2}}{13 b c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*Sqrt[a*c + b*c*x],x]

[Out]

(2*(a*c + b*c*x)^(13/2))/(13*b*c^6)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x)^5 \sqrt {a c+b c x} \, dx &=\frac {\int (a c+b c x)^{11/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{13/2}}{13 b c^6}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.14 \begin {gather*} \frac {2 (a+b x)^6 \sqrt {c (a+b x)}}{13 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*Sqrt[a*c + b*c*x],x]

[Out]

(2*(a + b*x)^6*Sqrt[c*(a + b*x)])/(13*b)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.13, size = 81, normalized size = 3.68 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 b^{\frac {11}{2}} \sqrt {c} \left (\frac {a}{b}+x\right )^{\frac {13}{2}}}{13},\text {Abs}\left [\frac {a}{b}+x\right ]<1\right \}\right \},b^{\frac {11}{2}} \sqrt {c} \text {meijerg}\left [\left \{\left \{1\right \},\left \{\frac {15}{2}\right \}\right \},\left \{\left \{\frac {13}{2}\right \},\left \{0\right \}\right \},\frac {a}{b}+x\right ]+b^{\frac {11}{2}} \sqrt {c} \text {meijerg}\left [\left \{\left \{\frac {15}{2},1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{\frac {13}{2},0\right \}\right \},\frac {a}{b}+x\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^5*(a*c + b*c*x)^(1/2),x]')

[Out]

Piecewise[{{2 b ^ (11 / 2) Sqrt[c] (a / b + x) ^ (13 / 2) / 13, Abs[a / b + x] < 1}}, b ^ (11 / 2) Sqrt[c] mei
jerg[{{1}, {15 / 2}}, {{13 / 2}, {0}}, a / b + x] + b ^ (11 / 2) Sqrt[c] meijerg[{{15 / 2, 1}, {}}, {{}, {13 /
2, 0}}, a / b + x]]

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Maple [A]
time = 0.17, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(\frac {2 \left (b c x +a c \right )^{\frac {13}{2}}}{13 b \,c^{6}}\)	\(19\)
default	\(\frac {2 \left (b c x +a c \right )^{\frac {13}{2}}}{13 b \,c^{6}}\)	\(19\)
gosper	\(\frac {2 \left (b x +a \right )^{6} \sqrt {b c x +a c}}{13 b}\)	\(23\)
trager	\(\frac {2 \left (x^{6} b^{6}+6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+20 a^{3} b^{3} x^{3}+15 a^{4} x^{2} b^{2}+6 a^{5} x b +a^{6}\right ) \sqrt {b c x +a c}}{13 b}\)	\(76\)
risch	\(\frac {2 c \left (x^{6} b^{6}+6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+20 a^{3} b^{3} x^{3}+15 a^{4} x^{2} b^{2}+6 a^{5} x b +a^{6}\right ) \left (b x +a \right )}{13 b \sqrt {c \left (b x +a \right )}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/13*(b*c*x+a*c)^(13/2)/b/c^6

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Maxima [A]
time = 0.26, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {13}{2}}}{13 \, b c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(1/2),x, algorithm="maxima")

[Out]

2/13*(b*c*x + a*c)^(13/2)/(b*c^6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (18) = 36\).
time = 0.29, size = 75, normalized size = 3.41 \begin {gather*} \frac {2 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \sqrt {b c x + a c}}{13 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(1/2),x, algorithm="fricas")

[Out]

2/13*(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*sqrt(b*c*x +
a*c)/b

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Sympy [A]
time = 0.66, size = 66, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {2 b^{\frac {11}{2}} \sqrt {c} \left (\frac {a}{b} + x\right )^{\frac {13}{2}}}{13} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {15}{2} \\\frac {13}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} + b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {15}{2}, 1 & \\ & \frac {13}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(b*c*x+a*c)**(1/2),x)

[Out]

Piecewise((2*b**(11/2)*sqrt(c)*(a/b + x)**(13/2)/13, Abs(a/b + x) < 1), (b**(11/2)*sqrt(c)*meijerg(((1,), (15/
2,)), ((13/2,), (0,)), a/b + x) + b**(11/2)*sqrt(c)*meijerg(((15/2, 1), ()), ((), (13/2, 0)), a/b + x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (18) = 36\).
time = 0.00, size = 820, normalized size = 37.27 \begin {gather*} \frac {\frac {2 b^{6} \left (\frac {1}{13} \sqrt {a c+b c x} \left (a c+b c x\right )^{6}-\frac {6}{11} \sqrt {a c+b c x} \left (a c+b c x\right )^{5} a c+\frac {5}{3} \sqrt {a c+b c x} \left (a c+b c x\right )^{4} a^{2} c^{2}-\frac {20}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a^{3} c^{3}+3 \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{4} c^{4}-2 \sqrt {a c+b c x} \left (a c+b c x\right ) a^{5} c^{5}+\sqrt {a c+b c x} a^{6} c^{6}\right )}{c^{6} b^{6}}+\frac {12 a b^{5} \left (\frac {1}{11} \sqrt {a c+b c x} \left (a c+b c x\right )^{5}-\frac {5}{9} \sqrt {a c+b c x} \left (a c+b c x\right )^{4} a c+\frac {10}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a^{2} c^{2}-2 \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{3} c^{3}+\frac {5}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a^{4} c^{4}-\sqrt {a c+b c x} a^{5} c^{5}\right )}{c^{5} b^{5}}+\frac {30 a^{2} b^{4} \left (\frac {1}{9} \sqrt {a c+b c x} \left (a c+b c x\right )^{4}-\frac {4}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3} a c+\frac {6}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a^{2} c^{2}-\frac {4}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a^{3} c^{3}+\sqrt {a c+b c x} a^{4} c^{4}\right )}{c^{4} b^{4}}+\frac {40 a^{3} b^{3} \left (\frac {1}{7} \sqrt {a c+b c x} \left (a c+b c x\right )^{3}-\frac {3}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2} a c+\sqrt {a c+b c x} \left (a c+b c x\right ) a^{2} c^{2}-\sqrt {a c+b c x} a^{3} c^{3}\right )}{c^{3} b^{3}}+\frac {30 a^{4} b^{2} \left (\frac {1}{5} \sqrt {a c+b c x} \left (a c+b c x\right )^{2}-\frac {2}{3} \sqrt {a c+b c x} \left (a c+b c x\right ) a c+\sqrt {a c+b c x} a^{2} c^{2}\right )}{c^{2} b^{2}}+2 a^{6} \sqrt {a c+b c x}+\frac {12 a^{5} \left (\frac {1}{3} \sqrt {a c+b c x} \left (a c+b c x\right )-a c \sqrt {a c+b c x}\right )}{c}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(1/2),x)

[Out]

2/3003*(3003*sqrt(b*c*x + a*c)*a^6 - 6006*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^5/c + 3003*(15*sqr
t(b*c*x + a*c)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c + 3*(b*c*x + a*c)^(5/2))*a^4/c^2 - 1716*(35*sqrt(b*c*x + a
*c)*a^3*c^3 - 35*(b*c*x + a*c)^(3/2)*a^2*c^2 + 21*(b*c*x + a*c)^(5/2)*a*c - 5*(b*c*x + a*c)^(7/2))*a^3/c^3 + 1
43*(315*sqrt(b*c*x + a*c)*a^4*c^4 - 420*(b*c*x + a*c)^(3/2)*a^3*c^3 + 378*(b*c*x + a*c)^(5/2)*a^2*c^2 - 180*(b
*c*x + a*c)^(7/2)*a*c + 35*(b*c*x + a*c)^(9/2))*a^2/c^4 - 26*(693*sqrt(b*c*x + a*c)*a^5*c^5 - 1155*(b*c*x + a*
c)^(3/2)*a^4*c^4 + 1386*(b*c*x + a*c)^(5/2)*a^3*c^3 - 990*(b*c*x + a*c)^(7/2)*a^2*c^2 + 385*(b*c*x + a*c)^(9/2
)*a*c - 63*(b*c*x + a*c)^(11/2))*a/c^5 + (3003*sqrt(b*c*x + a*c)*a^6*c^6 - 6006*(b*c*x + a*c)^(3/2)*a^5*c^5 +
9009*(b*c*x + a*c)^(5/2)*a^4*c^4 - 8580*(b*c*x + a*c)^(7/2)*a^3*c^3 + 5005*(b*c*x + a*c)^(9/2)*a^2*c^2 - 1638*
(b*c*x + a*c)^(11/2)*a*c + 231*(b*c*x + a*c)^(13/2))/c^6)/b

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Mupad [B]
time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} \frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{13/2}}{13\,b\,c^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + b*c*x)^(1/2)*(a + b*x)^5,x)

[Out]

(2*(c*(a + b*x))^(13/2))/(13*b*c^6)

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