3.15.46 \(\int \frac {(a+b x)^5}{\sqrt {a c+b c x}} \, dx\) [1446]

Optimal. Leaf size=22 \[ \frac {2 (a c+b c x)^{11/2}}{11 b c^6} \]

[Out]

2/11*(b*c*x+a*c)^(11/2)/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)^{11/2}}{11 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)^(11/2))/(11*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}{\sqrt {a c+b c x}} \, dx &=\frac {\int (a c+b c x)^{9/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{11/2}}{11 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}{11 b \sqrt {c (a+b x)}} \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)/(11*b*Sqrt[c*(a + b*x)])

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

method	result	size
derivativedivides	\(\frac {2 \left (b c x +a c \right )^{\frac {11}{2}}}{11 b \,c^{6}}\)	\(19\)
default	\(\frac {2 \left (b c x +a c \right )^{\frac {11}{2}}}{11 b \,c^{6}}\)	\(19\)
gosper	\(\frac {2 \left (b x +a \right )^{6}}{11 b \sqrt {b c x +a c}}\)	\(23\)
trager	\(\frac {2 \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right ) \sqrt {b c x +a c}}{11 c b}\)	\(68\)
risch	\(\frac {2 \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right ) \left (b x +a \right )}{11 b \sqrt {c \left (b x +a \right )}}\)	\(69\)

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]

2/11*(b*c*x+a*c)^(11/2)/b/c^6

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (18) = 36\).
time = 0.29, size = 374, normalized size = 17.00 \begin {gather*} \frac {2 \, {\left (693 \, \sqrt {b c x + a c} a^{5} - \frac {1155 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{4}}{c} + \frac {462 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{3}}{c^{2}} - \frac {198 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{2}}{c^{3}} + \frac {11 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a}{c^{4}} - \frac {693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}}{c^{5}}\right )}}{693 \, b c} \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]

2/693*(693*sqrt(b*c*x + a*c)*a^5 - 1155*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^4/c + 462*(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^3/c^2 - 198*(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^2/c^3 + 11*(3
15*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/c^4 - (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))/c^5)/(b*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (18) = 36\).
time = 0.74, size = 67, normalized size = 3.05 \begin {gather*} \frac {2 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {b c x + a c}}{11 \, b c} \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]

2/11*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)*sqrt(b*c*x + a*c)/(b*c)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
time = 0.48, 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 {9}{2}} \left (\frac {a}{b} + x\right )^{\frac {11}{2}}}{11 \sqrt {c}} & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \vee \left |{\frac {a}{b} + x}\right | < 1 \\\frac {b^{\frac {9}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {13}{2} \\\frac {11}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{\sqrt {c}} + \frac {b^{\frac {9}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {13}{2}, 1 & \\ & \frac {11}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )}}{\sqrt {c}} & \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((0, (Abs(a/b + x) < 1) & (1/Abs(a/b + x) < 1)), (2*b**(9/2)*(a/b + x)**(11/2)/(11*sqrt(c)), (Abs(a/b
+ x) < 1) | (1/Abs(a/b + x) < 1)), (b**(9/2)*meijerg(((1,), (13/2,)), ((11/2,), (0,)), a/b + x)/sqrt(c) + b**
(9/2)*meijerg(((13/2, 1), ()), ((), (11/2, 0)), a/b + x)/sqrt(c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (18) = 36\).
time = 0.67, size = 374, normalized size = 17.00 \begin {gather*} \frac {2 \, {\left (693 \, \sqrt {b c x + a c} a^{5} - \frac {1155 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{4}}{c} + \frac {462 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{3}}{c^{2}} - \frac {198 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{2}}{c^{3}} + \frac {11 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a}{c^{4}} - \frac {693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}}{c^{5}}\right )}}{693 \, b c} \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="giac")

[Out]

2/693*(693*sqrt(b*c*x + a*c)*a^5 - 1155*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^4/c + 462*(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^3/c^2 - 198*(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^2/c^3 + 11*(3
15*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/c^4 - (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))/c^5)/(b*c)

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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 )}^{11/2}}{11\,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)^(1/2),x)

[Out]

(2*(c*(a + b*x))^(11/2))/(11*b*c^6)

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