Optimal. Leaf size=131 \[ -2 A b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {794, 664, 660, 207} \[ \frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}-2 A b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 664
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx &=\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+A \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx\\ &=\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+(A b) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx\\ &=\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (A b^2\right ) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (A b^3\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (2 A b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}-2 A b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 95, normalized size = 0.73 \[ \frac {(x (b+c x))^{5/2} \left (-\frac {14 A b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{(b+c x)^{5/2}}+A \left (\frac {14 b^2}{(b+c x)^2}+\frac {14 b}{3 (b+c x)}+\frac {14}{5}\right )+\frac {2 B (b+c x)}{c}\right )}{7 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 245, normalized size = 1.87 \[ \left [\frac {105 \, A b^{\frac {5}{2}} c x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \, {\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} + {\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{105 \, c x}, \frac {2 \, {\left (105 \, A \sqrt {-b} b^{2} c x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \, {\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} + {\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{105 \, c x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 139, normalized size = 1.06 \[ \frac {2 \, A b^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, {\left (105 \, A b^{3} c \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 15 \, B \sqrt {-b} b^{\frac {7}{2}} + 161 \, A \sqrt {-b} b^{\frac {5}{2}} c\right )}}{105 \, \sqrt {-b} c} + \frac {2 \, {\left (15 \, {\left (c x + b\right )}^{\frac {7}{2}} B c^{6} + 21 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{7} + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{7} + 105 \, \sqrt {c x + b} A b^{2} c^{7}\right )}}{105 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 151, normalized size = 1.15 \[ -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (-15 \sqrt {c x +b}\, B \,c^{3} x^{3}-21 \sqrt {c x +b}\, A \,c^{3} x^{2}-45 \sqrt {c x +b}\, B b \,c^{2} x^{2}+105 A \,b^{\frac {5}{2}} c \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-77 \sqrt {c x +b}\, A b \,c^{2} x -45 \sqrt {c x +b}\, B \,b^{2} c x -161 \sqrt {c x +b}\, A \,b^{2} c -15 \sqrt {c x +b}\, B \,b^{3}\right )}{105 \sqrt {c x +b}\, c \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ A b^{2} \int \frac {\sqrt {c x + b}}{x}\,{d x} + \frac {2 \, {\left (35 \, {\left (B b^{2} c + 2 \, A b c^{2}\right )} x^{3} + {\left (15 \, B c^{3} x^{3} + 3 \, B b c^{2} x^{2} - 4 \, B b^{2} c x + 8 \, B b^{3}\right )} x^{2} + 35 \, {\left (B b^{3} + 2 \, A b^{2} c\right )} x^{2} + 7 \, {\left (3 \, {\left (2 \, B b c^{2} + A c^{3}\right )} x^{3} + {\left (2 \, B b^{2} c + A b c^{2}\right )} x^{2} - 2 \, {\left (2 \, B b^{3} + A b^{2} c\right )} x\right )} x\right )} \sqrt {c x + b}}{105 \, c x^{2}} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^{7/2}} \,d x \]
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 \[ \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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