Optimal. Leaf size=153 \[ -\frac {5}{8} (b B+6 A c) (b+2 c x) \sqrt {b x+c x^2}-\frac {5 c (b B+6 A c) \left (b x+c x^2\right )^{3/2}}{3 b}+\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}+\frac {5 b^2 (b B+6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 \sqrt {c}} \]
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Rubi [A]
time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {806, 676, 678,
626, 634, 212} \begin {gather*} \frac {5 b^2 (6 A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 \sqrt {c}}+\frac {2 \left (b x+c x^2\right )^{5/2} (6 A c+b B)}{b x^2}-\frac {5 c \left (b x+c x^2\right )^{3/2} (6 A c+b B)}{3 b}-\frac {5}{8} (b+2 c x) \sqrt {b x+c x^2} (6 A c+b B)-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 676
Rule 678
Rule 806
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^4} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}+\frac {\left (2 \left (-4 (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^3} \, dx}{b}\\ &=\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}-\frac {(5 c (b B+6 A c)) \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx}{b}\\ &=-\frac {5 c (b B+6 A c) \left (b x+c x^2\right )^{3/2}}{3 b}+\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}-\frac {1}{2} (5 c (b B+6 A c)) \int \sqrt {b x+c x^2} \, dx\\ &=-\frac {5}{8} (b B+6 A c) (b+2 c x) \sqrt {b x+c x^2}-\frac {5 c (b B+6 A c) \left (b x+c x^2\right )^{3/2}}{3 b}+\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}+\frac {1}{16} \left (5 b^2 (b B+6 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {5}{8} (b B+6 A c) (b+2 c x) \sqrt {b x+c x^2}-\frac {5 c (b B+6 A c) \left (b x+c x^2\right )^{3/2}}{3 b}+\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}+\frac {1}{8} \left (5 b^2 (b B+6 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {5}{8} (b B+6 A c) (b+2 c x) \sqrt {b x+c x^2}-\frac {5 c (b B+6 A c) \left (b x+c x^2\right )^{3/2}}{3 b}+\frac {2 (b B+6 A c) \left (b x+c x^2\right )^{5/2}}{b x^2}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{b x^4}+\frac {5 b^2 (b B+6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 118, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (-6 A \left (8 b^2-9 b c x-2 c^2 x^2\right )+B x \left (33 b^2+26 b c x+8 c^2 x^2\right )-\frac {15 b^2 (b B+6 A c) \sqrt {x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {c} \sqrt {b+c x}}\right )}{24 x} \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. \(341\) vs.
\(2(133)=266\).
time = 0.53, size = 342, normalized size = 2.24
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (-8 B \,c^{2} x^{3}-12 A \,c^{2} x^{2}-26 b B \,x^{2} c -54 A b c x -33 b^{2} B x +48 b^{2} A \right )}{24 \sqrt {x \left (c x +b \right )}}+\frac {15 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) \sqrt {c}\, A}{8}+\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B}{16 \sqrt {c}}\) | \(130\) |
default | \(B \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{b \,x^{3}}-\frac {8 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{3 b \,x^{2}}-\frac {10 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5}+\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2}\right )}{3 b}\right )}{b}\right )+A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{b \,x^{4}}+\frac {6 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{b \,x^{3}}-\frac {8 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{3 b \,x^{2}}-\frac {10 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5}+\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2}\right )}{3 b}\right )}{b}\right )}{b}\right )\) | \(342\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 172, normalized size = 1.12 \begin {gather*} \frac {5 \, B b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, \sqrt {c}} + \frac {15}{8} \, A b^{2} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + \frac {5}{8} \, \sqrt {c x^{2} + b x} B b^{2} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{12 \, x} - \frac {15 \, \sqrt {c x^{2} + b x} A b^{2}}{4 \, x} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{3 \, x^{2}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{4 \, x^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.39, size = 240, normalized size = 1.57 \begin {gather*} \left [\frac {15 \, {\left (B b^{3} + 6 \, A b^{2} c\right )} \sqrt {c} x \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, B c^{3} x^{3} - 48 \, A b^{2} c + 2 \, {\left (13 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2} + 3 \, {\left (11 \, B b^{2} c + 18 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c x}, -\frac {15 \, {\left (B b^{3} + 6 \, A b^{2} c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8 \, B c^{3} x^{3} - 48 \, A b^{2} c + 2 \, {\left (13 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2} + 3 \, {\left (11 \, B b^{2} c + 18 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c x}\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 {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 141, normalized size = 0.92 \begin {gather*} \frac {2 \, A b^{3}}{\sqrt {c} x - \sqrt {c x^{2} + b x}} + \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, B c^{2} x + \frac {13 \, B b c^{3} + 6 \, A c^{4}}{c^{2}}\right )} x + \frac {3 \, {\left (11 \, B b^{2} c^{2} + 18 \, A b c^{3}\right )}}{c^{2}}\right )} - \frac {5 \, {\left (B b^{3} + 6 \, A b^{2} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, \sqrt {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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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