Optimal. Leaf size=170 \[ -\frac {b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{7/2}}+\frac {b^3 (b+2 c x) \sqrt {b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac {b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x} \]
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Rubi [A] time = 0.13, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {794, 664, 612, 620, 206} \begin {gather*} \frac {b^3 (b+2 c x) \sqrt {b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac {b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{7/2}}-\frac {b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 664
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x} \, dx &=\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}+\frac {\left (b B-A c+\frac {7}{2} (-b B+2 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x} \, dx}{6 c}\\ &=-\frac {(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac {(b (5 b B-12 A c)) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c}\\ &=-\frac {b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}+\frac {\left (b^3 (5 b B-12 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^2}\\ &=\frac {b^3 (5 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^3}-\frac {b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac {\left (b^5 (5 b B-12 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^3}\\ &=\frac {b^3 (5 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^3}-\frac {b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac {\left (b^5 (5 b B-12 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{512 c^3}\\ &=\frac {b^3 (5 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^3}-\frac {b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac {B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac {b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 166, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-10 b^4 c (18 A+5 B x)+40 b^3 c^2 x (3 A+B x)+48 b^2 c^3 x^2 (62 A+45 B x)+64 b c^4 x^3 (63 A+50 B x)+256 c^5 x^4 (6 A+5 B x)+75 b^5 B\right )-\frac {15 b^{9/2} (5 b B-12 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{7680 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.89, size = 177, normalized size = 1.04 \begin {gather*} \frac {\left (5 b^6 B-12 A b^5 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{1024 c^{7/2}}+\frac {\sqrt {b x+c x^2} \left (-180 A b^4 c+120 A b^3 c^2 x+2976 A b^2 c^3 x^2+4032 A b c^4 x^3+1536 A c^5 x^4+75 b^5 B-50 b^4 B c x+40 b^3 B c^2 x^2+2160 b^2 B c^3 x^3+3200 b B c^4 x^4+1280 B c^5 x^5\right )}{7680 c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 351, normalized size = 2.06 \begin {gather*} \left [-\frac {15 \, {\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (1280 \, B c^{6} x^{5} + 75 \, B b^{5} c - 180 \, A b^{4} c^{2} + 128 \, {\left (25 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 144 \, {\left (15 \, B b^{2} c^{4} + 28 \, A b c^{5}\right )} x^{3} + 8 \, {\left (5 \, B b^{3} c^{3} + 372 \, A b^{2} c^{4}\right )} x^{2} - 10 \, {\left (5 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{4}}, \frac {15 \, {\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (1280 \, B c^{6} x^{5} + 75 \, B b^{5} c - 180 \, A b^{4} c^{2} + 128 \, {\left (25 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 144 \, {\left (15 \, B b^{2} c^{4} + 28 \, A b c^{5}\right )} x^{3} + 8 \, {\left (5 \, B b^{3} c^{3} + 372 \, A b^{2} c^{4}\right )} x^{2} - 10 \, {\left (5 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 198, normalized size = 1.16 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c^{2} x + \frac {25 \, B b c^{6} + 12 \, A c^{7}}{c^{5}}\right )} x + \frac {9 \, {\left (15 \, B b^{2} c^{5} + 28 \, A b c^{6}\right )}}{c^{5}}\right )} x + \frac {5 \, B b^{3} c^{4} + 372 \, A b^{2} c^{5}}{c^{5}}\right )} x - \frac {5 \, {\left (5 \, B b^{4} c^{3} - 12 \, A b^{3} c^{4}\right )}}{c^{5}}\right )} x + \frac {15 \, {\left (5 \, B b^{5} c^{2} - 12 \, A b^{4} c^{3}\right )}}{c^{5}}\right )} + \frac {{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 274, normalized size = 1.61 \begin {gather*} \frac {3 A \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {5}{2}}}-\frac {5 B \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {7}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} x}{64 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{4} x}{256 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{4}}{128 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b x}{8}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{5}}{512 c^{3}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} x}{96 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2}}{16 c}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3}}{192 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B x}{6}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A}{5}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b}{12 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 271, normalized size = 1.59 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B x + \frac {1}{8} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b x + \frac {5 \, \sqrt {c x^{2} + b x} B b^{4} x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} x}{96 \, c} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} x}{64 \, c} - \frac {5 \, B b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} + \frac {3 \, A b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {5}{2}}} + \frac {1}{5} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A + \frac {5 \, \sqrt {c x^{2} + b x} B b^{5}}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3}}{192 \, c^{2}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{4}}{128 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b}{12 \, c} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2}}{16 \, c} \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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x} \,d x \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 {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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