Optimal. Leaf size=211 \[ \frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\sqrt {c} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2}}+\frac {d^{3/2} (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.13, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {99, 154, 159,
163, 65, 223, 212, 95, 214} \begin {gather*} \frac {\sqrt {c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2}}+\frac {d^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{4 a x}+\frac {d \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^3} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {1}{2} \int \frac {(c+d x)^{3/2} \left (\frac {1}{2} (b c+5 a d)+3 b d x\right )}{x^2 \sqrt {a+b x}} \, dx\\ &=-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{4} \left (-b^2 c^2+10 a b c d+15 a^2 d^2\right )+\frac {1}{2} b d (b c+11 a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 a}\\ &=\frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {-\frac {1}{4} b c \left (b^2 c^2-10 a b c d-15 a^2 d^2\right )+a b d^2 (5 b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a b}\\ &=\frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {1}{2} \left (d^2 (5 b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (c \left (b^2 c^2-10 a b c d-15 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a}\\ &=\frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\left (d^2 (5 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b}-\frac {\left (c \left (b^2 c^2-10 a b c d-15 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a}\\ &=\frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\sqrt {c} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2}}+\frac {\left (d^2 (5 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b}\\ &=\frac {d (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{2 x^2}+\frac {\sqrt {c} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2}}+\frac {d^{3/2} (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 177, normalized size = 0.84 \begin {gather*} \frac {\sqrt {c} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-2 a c^2-b c^2 x-9 a c d x+4 a d^2 x^2\right )}{x^2}+\frac {4 a d^{3/2} (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}\right )}{4 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs.
\(2(167)=334\).
time = 0.07, size = 442, normalized size = 2.09
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x^{2} \sqrt {b d}+10 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a b \,c^{2} d \,x^{2} \sqrt {b d}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{2} c^{3} x^{2} \sqrt {b d}-4 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x^{2} \sqrt {a c}-20 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x^{2} \sqrt {a c}-8 a \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+18 a c d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+2 b \,c^{2} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+4 a \,c^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\right )}{8 a \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(442\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.47, size = 1094, normalized size = 5.18 \begin {gather*} \left [\frac {4 \, {\left (5 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left (b^{2} c^{2} - 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (4 \, a d^{2} x^{2} - 2 \, a c^{2} - {\left (b c^{2} + 9 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a x^{2}}, -\frac {8 \, {\left (5 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b^{2} c^{2} - 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (4 \, a d^{2} x^{2} - 2 \, a c^{2} - {\left (b c^{2} + 9 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a x^{2}}, -\frac {{\left (b^{2} c^{2} - 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (5 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 2 \, {\left (4 \, a d^{2} x^{2} - 2 \, a c^{2} - {\left (b c^{2} + 9 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a x^{2}}, -\frac {{\left (b^{2} c^{2} - 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 4 \, {\left (5 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (4 \, a d^{2} x^{2} - 2 \, a c^{2} - {\left (b c^{2} + 9 \, a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1200 vs.
\(2 (167) = 334\).
time = 2.62, size = 1200, normalized size = 5.69 \begin {gather*} \frac {\frac {4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d^{2} {\left | b \right |}}{b} - \frac {2 \, {\left (5 \, \sqrt {b d} b c d {\left | b \right |} + \sqrt {b d} a d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} + \frac {{\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 10 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} - 15 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} - \frac {2 \, {\left (\sqrt {b d} b^{9} c^{6} {\left | b \right |} + 5 \, \sqrt {b d} a b^{8} c^{5} d {\left | b \right |} - 30 \, \sqrt {b d} a^{2} b^{7} c^{4} d^{2} {\left | b \right |} + 50 \, \sqrt {b d} a^{3} b^{6} c^{3} d^{3} {\left | b \right |} - 35 \, \sqrt {b d} a^{4} b^{5} c^{2} d^{4} {\left | b \right |} + 9 \, \sqrt {b d} a^{5} b^{4} c d^{5} {\left | b \right |} - 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{5} {\left | b \right |} - 20 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{4} d {\left | b \right |} + 22 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{3} d^{2} {\left | b \right |} + 28 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c^{2} d^{3} {\left | b \right |} - 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} c d^{4} {\left | b \right |} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{4} {\left | b \right |} + 29 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{3} d {\left | b \right |} + 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c^{2} d^{2} {\left | b \right |} + 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} c d^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{3} c^{3} {\left | b \right |} - 14 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{2} c^{2} d {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b c d^{2} {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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