Optimal. Leaf size=97 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{7/2}}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}-\frac{2 x^3}{c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0398188, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {668, 670, 640, 620, 206} \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{7/2}}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}-\frac{2 x^3}{c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 668
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 x^3}{c \sqrt{b x+c x^2}}+\frac{5 \int \frac{x^2}{\sqrt{b x+c x^2}} \, dx}{c}\\ &=-\frac{2 x^3}{c \sqrt{b x+c x^2}}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{(15 b) \int \frac{x}{\sqrt{b x+c x^2}} \, dx}{4 c^2}\\ &=-\frac{2 x^3}{c \sqrt{b x+c x^2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}+\frac{\left (15 b^2\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{8 c^3}\\ &=-\frac{2 x^3}{c \sqrt{b x+c x^2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{4 c^3}\\ &=-\frac{2 x^3}{c \sqrt{b x+c x^2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}+\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0123995, size = 50, normalized size = 0.52 \[ \frac{2 x^4 \sqrt{\frac{c x}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{c x}{b}\right )}{7 b \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 93, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{5\,b{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{15\,{b}^{2}x}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{15\,{b}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08516, size = 410, normalized size = 4.23 \begin{align*} \left [\frac{15 \,{\left (b^{2} c x + b^{3}\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (2 \, c^{3} x^{2} - 5 \, b c^{2} x - 15 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{8 \,{\left (c^{5} x + b c^{4}\right )}}, -\frac{15 \,{\left (b^{2} c x + b^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (2 \, c^{3} x^{2} - 5 \, b c^{2} x - 15 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{4 \,{\left (c^{5} x + b c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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