3.15.47 \(\int \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx\) [1447]

Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{9/2}}{9 b c^6} \]

[Out]

2/9*(b*c*x+a*c)^(9/2)/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)^{9/2}}{9 b c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^(3/2),x]

[Out]

(2*(a*c + b*c*x)^(9/2))/(9*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 \frac {(a+b x)^5}{(a c+b c x)^{3/2}} \, dx &=\frac {\int (a c+b c x)^{7/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{9/2}}{9 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}{9 b (c (a+b x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^6)/(9*b*(c*(a + b*x))^(3/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.67, size = 119, normalized size = 5.41 \begin {gather*} \text {Piecewise}\left [\left \{\left \{0,\text {Abs}\left [\frac {a}{b}+x\right ]<1\text {\&\&}\text {Abs}\left [\frac {b}{a+b x}\right ]<1\right \},\left \{\frac {2 b^{\frac {7}{2}} \left (\frac {a}{b}+x\right )^{\frac {9}{2}}}{9 c^{\frac {3}{2}}},\text {Abs}\left [\frac {a}{b}+x\right ]<1\text {$\vert $$\vert $}\text {Abs}\left [\frac {b}{a+b x}\right ]<1\right \}\right \},\frac {b^{\frac {7}{2}} \text {meijerg}\left [\left \{\left \{1\right \},\left \{\frac {11}{2}\right \}\right \},\left \{\left \{\frac {9}{2}\right \},\left \{0\right \}\right \},\frac {a}{b}+x\right ]}{c^{\frac {3}{2}}}+\frac {b^{\frac {7}{2}} \text {meijerg}\left [\left \{\left \{\frac {11}{2},1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{\frac {9}{2},0\right \}\right \},\frac {a}{b}+x\right ]}{c^{\frac {3}{2}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{0, Abs[a / b + x] < 1 && Abs[b / (a + b x)] < 1}, {2 b ^ (7 / 2) (a / b + x) ^ (9 / 2) / (9 c ^ (3
/ 2)), Abs[a / b + x] < 1 || Abs[b / (a + b x)] < 1}}, b ^ (7 / 2) meijerg[{{1}, {11 / 2}}, {{9 / 2}, {0}}, a
/ b + x] / c ^ (3 / 2) + b ^ (7 / 2) meijerg[{{11 / 2, 1}, {}}, {{}, {9 / 2, 0}}, a / b + x] / c ^ (3 / 2)]

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Maple [A]
time = 0.16, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(\frac {2 \left (b c x +a c \right )^{\frac {9}{2}}}{9 b \,c^{6}}\)	\(19\)
default	\(\frac {2 \left (b c x +a c \right )^{\frac {9}{2}}}{9 b \,c^{6}}\)	\(19\)
gosper	\(\frac {2 \left (b x +a \right )^{6}}{9 b \left (b c x +a c \right )^{\frac {3}{2}}}\)	\(23\)
trager	\(\frac {2 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \sqrt {b c x +a c}}{9 c^{2} b}\)	\(57\)
risch	\(\frac {2 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b x +a \right )}{9 c b \sqrt {c \left (b x +a \right )}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*c*x+a*c)^(9/2)/b/c^6

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Maxima [A]
time = 0.27, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b c x + a c\right )}^{\frac {9}{2}}}{9 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/9*(b*c*x + a*c)^(9/2)/(b*c^6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
time = 0.30, size = 56, normalized size = 2.55 \begin {gather*} \frac {2 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {b c x + a c}}{9 \, b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(b*c*x + a*c)/(b*c^2)

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Sympy [A]
time = 0.60, size = 88, normalized size = 4.00 \begin {gather*} \begin {cases} 0 & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \wedge \left |{\frac {a}{b} + x}\right | < 1 \\\frac {2 b^{\frac {7}{2}} \left (\frac {a}{b} + x\right )^{\frac {9}{2}}}{9 c^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \vee \left |{\frac {a}{b} + x}\right | < 1 \\\frac {b^{\frac {7}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {11}{2} \\\frac {9}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {3}{2}}} + \frac {b^{\frac {7}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {11}{2}, 1 & \\ & \frac {9}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{c^{\frac {3}{2}}} & \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)**(3/2),x)

[Out]

Piecewise((0, (Abs(a/b + x) < 1) & (1/Abs(a/b + x) < 1)), (2*b**(7/2)*(a/b + x)**(9/2)/(9*c**(3/2)), (Abs(a/b
+ x) < 1) | (1/Abs(a/b + x) < 1)), (b**(7/2)*meijerg(((1,), (11/2,)), ((9/2,), (0,)), a/b + x)/c**(3/2) + b**(
7/2)*meijerg(((11/2, 1), ()), ((), (9/2, 0)), a/b + x)/c**(3/2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (18) = 36\).
time = 0.00, size = 426, normalized size = 19.36 \begin {gather*} \frac {\frac {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 {8 a 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 {12 a^{2} 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^{4} \sqrt {a c+b c x}+\frac {8 a^{3} \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}}{c^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(3/2),x)

[Out]

2/315*(315*sqrt(b*c*x + a*c)*a^4 - 420*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^3/c + 126*(15*sqrt(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^2/c^2 - 36*(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/c^3 + (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))/c^4)/(b*c^2)

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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 )}^{9/2}}{9\,b\,c^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a*c + b*c*x)^(3/2),x)

[Out]

(2*(c*(a + b*x))^(9/2))/(9*b*c^6)

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