Optimal. Leaf size=128 \[ \frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}-\frac {35 b^3 \sqrt {b x+c x^2}}{64 c^4}+\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c} \]
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Rubi [A] time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {670, 640, 620, 206} \[ -\frac {35 b^3 \sqrt {b x+c x^2}}{64 c^4}+\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}+\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {b x+c x^2}} \, dx &=\frac {x^3 \sqrt {b x+c x^2}}{4 c}-\frac {(7 b) \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx}{8 c}\\ &=-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (35 b^2\right ) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{48 c^2}\\ &=\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c}-\frac {\left (35 b^3\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{64 c^3}\\ &=-\frac {35 b^3 \sqrt {b x+c x^2}}{64 c^4}+\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (35 b^4\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^4}\\ &=-\frac {35 b^3 \sqrt {b x+c x^2}}{64 c^4}+\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (35 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^4}\\ &=-\frac {35 b^3 \sqrt {b x+c x^2}}{64 c^4}+\frac {35 b^2 x \sqrt {b x+c x^2}}{96 c^3}-\frac {7 b x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {x^3 \sqrt {b x+c x^2}}{4 c}+\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 98, normalized size = 0.77 \[ \frac {\sqrt {x (b+c x)} \left (\frac {105 b^{7/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (-105 b^3+70 b^2 c x-56 b c^2 x^2+48 c^3 x^3\right )\right )}{192 c^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 170, normalized size = 1.33 \[ \left [\frac {105 \, b^{4} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (48 \, c^{4} x^{3} - 56 \, b c^{3} x^{2} + 70 \, b^{2} c^{2} x - 105 \, b^{3} c\right )} \sqrt {c x^{2} + b x}}{384 \, c^{5}}, -\frac {105 \, b^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (48 \, c^{4} x^{3} - 56 \, b c^{3} x^{2} + 70 \, b^{2} c^{2} x - 105 \, b^{3} c\right )} \sqrt {c x^{2} + b x}}{192 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 89, normalized size = 0.70 \[ \frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, x {\left (\frac {6 \, x}{c} - \frac {7 \, b}{c^{2}}\right )} + \frac {35 \, b^{2}}{c^{3}}\right )} x - \frac {105 \, b^{3}}{c^{4}}\right )} - \frac {35 \, b^{4} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 112, normalized size = 0.88 \[ \frac {\sqrt {c \,x^{2}+b x}\, x^{3}}{4 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, b \,x^{2}}{24 c^{2}}+\frac {35 b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}+\frac {35 \sqrt {c \,x^{2}+b x}\, b^{2} x}{96 c^{3}}-\frac {35 \sqrt {c \,x^{2}+b x}\, b^{3}}{64 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 110, normalized size = 0.86 \[ \frac {\sqrt {c x^{2} + b x} x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b x^{2}}{24 \, c^{2}} + \frac {35 \, \sqrt {c x^{2} + b x} b^{2} x}{96 \, c^{3}} + \frac {35 \, b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} b^{3}}{64 \, c^{4}} \]
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 \[ \int \frac {x^4}{\sqrt {c\,x^2+b\,x}} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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