Optimal. Leaf size=119 \[ -\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {792, 662, 620, 206} \begin {gather*} -\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^7} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+B \int \frac {\left (b x+c x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+(B c) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^2\right ) \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^3\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (2 B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 83, normalized size = 0.70 \begin {gather*} \frac {2 (x (b+c x))^{5/2} \left ((b+c x)^3 (b B-A c)-\frac {b^4 B \, _2F_1\left (-\frac {7}{2},-\frac {7}{2};-\frac {5}{2};-\frac {c x}{b}\right )}{\sqrt {\frac {c x}{b}+1}}\right )}{7 b c x^6 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 120, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (15 A b^3+45 A b^2 c x+45 A b c^2 x^2+15 A c^3 x^3+21 b^3 B x+77 b^2 B c x^2+161 b B c^2 x^3\right )}{105 b x^4}-B c^{5/2} \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 238, normalized size = 2.00 \begin {gather*} \left [\frac {105 \, B b c^{\frac {5}{2}} x^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (15 \, A b^{3} + {\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} + {\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, b x^{4}}, -\frac {2 \, {\left (105 \, B b \sqrt {-c} c^{2} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (15 \, A b^{3} + {\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} + {\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{105 \, b x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 390, normalized size = 3.28 \begin {gather*} -B c^{\frac {5}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A c^{\frac {7}{2}} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{2} c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b c^{3} + 245 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{3} c^{\frac {3}{2}} + 525 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{4} c + 525 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} c^{2} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{5} \sqrt {c} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{4} c^{\frac {3}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{5} c + 15 \, A b^{6} \sqrt {c}\right )}}{105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 263, normalized size = 2.21 \begin {gather*} B \,c^{\frac {5}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )-\frac {4 \sqrt {c \,x^{2}+b x}\, B \,c^{4} x}{b^{2}}-\frac {2 \sqrt {c \,x^{2}+b x}\, B \,c^{3}}{b}+\frac {32 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,c^{5} x}{3 b^{4}}+\frac {16 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,c^{4}}{3 b^{3}}+\frac {256 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,c^{5}}{15 b^{5}}-\frac {256 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B \,c^{4}}{15 b^{5} x^{2}}+\frac {32 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B \,c^{3}}{5 b^{4} x^{3}}-\frac {16 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B \,c^{2}}{15 b^{3} x^{4}}-\frac {4 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B c}{15 b^{2} x^{5}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B}{5 b \,x^{6}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} A}{7 b \,x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.91, size = 258, normalized size = 2.17 \begin {gather*} B c^{\frac {5}{2}} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {38 \, \sqrt {c x^{2} + b x} B c^{2}}{15 \, x} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{7 \, b x} - \frac {7 \, \sqrt {c x^{2} + b x} B b c}{30 \, x^{2}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{7 \, x^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} B b^{2}}{10 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B c}{3 \, x^{3}} - \frac {3 \, \sqrt {c x^{2} + b x} A b c}{28 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{2 \, x^{4}} - \frac {15 \, \sqrt {c x^{2} + b x} A b^{2}}{28 \, x^{4}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{5 \, x^{5}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{4 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{x^{6}} \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^7} \,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^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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