Optimal. Leaf size=197 \[ -\frac {7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{11/2}}+\frac {7 b^3 \sqrt {b x+c x^2} (9 b B-10 A c)}{128 c^5}-\frac {7 b^2 x \sqrt {b x+c x^2} (9 b B-10 A c)}{192 c^4}+\frac {7 b x^2 \sqrt {b x+c x^2} (9 b B-10 A c)}{240 c^3}-\frac {x^3 \sqrt {b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.20, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {794, 670, 640, 620, 206} \[ \frac {7 b^3 \sqrt {b x+c x^2} (9 b B-10 A c)}{128 c^5}-\frac {7 b^2 x \sqrt {b x+c x^2} (9 b B-10 A c)}{192 c^4}-\frac {7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{11/2}}+\frac {7 b x^2 \sqrt {b x+c x^2} (9 b B-10 A c)}{240 c^3}-\frac {x^3 \sqrt {b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {B x^4 \sqrt {b x+c x^2}}{5 c}+\frac {\left (4 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {x^4}{\sqrt {b x+c x^2}} \, dx}{5 c}\\ &=-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}+\frac {(7 b (9 b B-10 A c)) \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx}{80 c^2}\\ &=\frac {7 b (9 b B-10 A c) x^2 \sqrt {b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}-\frac {\left (7 b^2 (9 b B-10 A c)\right ) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{96 c^3}\\ &=-\frac {7 b^2 (9 b B-10 A c) x \sqrt {b x+c x^2}}{192 c^4}+\frac {7 b (9 b B-10 A c) x^2 \sqrt {b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}+\frac {\left (7 b^3 (9 b B-10 A c)\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{128 c^4}\\ &=\frac {7 b^3 (9 b B-10 A c) \sqrt {b x+c x^2}}{128 c^5}-\frac {7 b^2 (9 b B-10 A c) x \sqrt {b x+c x^2}}{192 c^4}+\frac {7 b (9 b B-10 A c) x^2 \sqrt {b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}-\frac {\left (7 b^4 (9 b B-10 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^5}\\ &=\frac {7 b^3 (9 b B-10 A c) \sqrt {b x+c x^2}}{128 c^5}-\frac {7 b^2 (9 b B-10 A c) x \sqrt {b x+c x^2}}{192 c^4}+\frac {7 b (9 b B-10 A c) x^2 \sqrt {b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}-\frac {\left (7 b^4 (9 b B-10 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^5}\\ &=\frac {7 b^3 (9 b B-10 A c) \sqrt {b x+c x^2}}{128 c^5}-\frac {7 b^2 (9 b B-10 A c) x \sqrt {b x+c x^2}}{192 c^4}+\frac {7 b (9 b B-10 A c) x^2 \sqrt {b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {b x+c x^2}}{5 c}-\frac {7 b^4 (9 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 133, normalized size = 0.68 \[ \frac {\sqrt {x (b+c x)} \left (\frac {(9 b B-10 A c) \left (c x \sqrt {\frac {c x}{b}+1} \left (105 b^3-70 b^2 c x+56 b c^2 x^2-48 c^3 x^3\right )-105 b^{7/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{\sqrt {\frac {c x}{b}+1}}+384 B c^5 x^5\right )}{1920 c^6 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 303, normalized size = 1.54 \[ \left [-\frac {105 \, {\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c - 1050 \, A b^{3} c^{2} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3840 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c - 1050 \, A b^{3} c^{2} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{1920 \, c^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 165, normalized size = 0.84 \[ \frac {1}{1920} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, B x}{c} - \frac {9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac {7 \, {\left (9 \, B b^{2} c^{2} - 10 \, A b c^{3}\right )}}{c^{5}}\right )} x - \frac {35 \, {\left (9 \, B b^{3} c - 10 \, A b^{2} c^{2}\right )}}{c^{5}}\right )} x + \frac {105 \, {\left (9 \, B b^{4} - 10 \, A b^{3} c\right )}}{c^{5}}\right )} + \frac {7 \, {\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 255, normalized size = 1.29 \[ \frac {\sqrt {c \,x^{2}+b x}\, B \,x^{4}}{5 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,x^{3}}{4 c}-\frac {9 \sqrt {c \,x^{2}+b x}\, B b \,x^{3}}{40 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A b \,x^{2}}{24 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x}\, B \,b^{2} x^{2}}{80 c^{3}}+\frac {35 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}-\frac {63 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {11}{2}}}+\frac {35 \sqrt {c \,x^{2}+b x}\, A \,b^{2} x}{96 c^{3}}-\frac {21 \sqrt {c \,x^{2}+b x}\, B \,b^{3} x}{64 c^{4}}-\frac {35 \sqrt {c \,x^{2}+b x}\, A \,b^{3}}{64 c^{4}}+\frac {63 \sqrt {c \,x^{2}+b x}\, B \,b^{4}}{128 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 252, normalized size = 1.28 \[ \frac {\sqrt {c x^{2} + b x} B x^{4}}{5 \, c} - \frac {9 \, \sqrt {c x^{2} + b x} B b x^{3}}{40 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A x^{3}}{4 \, c} + \frac {21 \, \sqrt {c x^{2} + b x} B b^{2} x^{2}}{80 \, c^{3}} - \frac {7 \, \sqrt {c x^{2} + b x} A b x^{2}}{24 \, c^{2}} - \frac {21 \, \sqrt {c x^{2} + b x} B b^{3} x}{64 \, c^{4}} + \frac {35 \, \sqrt {c x^{2} + b x} A b^{2} x}{96 \, c^{3}} - \frac {63 \, B b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {11}{2}}} + \frac {35 \, A b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} + \frac {63 \, \sqrt {c x^{2} + b x} B b^{4}}{128 \, c^{5}} - \frac {35 \, \sqrt {c x^{2} + b x} A b^{3}}{64 \, c^{4}} \]
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 \[ \int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \]
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 \[ \int \frac {x^{4} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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