Integrand size = 26, antiderivative size = 96 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+16 c^{3/2} d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {700, 635, 212} \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=16 c^{3/2} d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 212
Rule 635
Rule 700
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (4 c d^2\right ) \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (16 c^2 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (32 c^2 d^4\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right ) \\ & = -\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+16 c^{3/2} d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=d^4 \left (-\frac {2 (b+2 c x) \left (b^2+16 b c x+4 c \left (3 a+4 c x^2\right )\right )}{3 (a+x (b+c x))^{3/2}}+32 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1136\) vs. \(2(82)=164\).
Time = 3.06 (sec) , antiderivative size = 1137, normalized size of antiderivative = 11.84
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (82) = 164\).
Time = 0.51 (sec) , antiderivative size = 440, normalized size of antiderivative = 4.58 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\left [\frac {2 \, {\left (12 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - {\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, -\frac {2 \, {\left (24 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.26 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-16 \, c^{\frac {3}{2}} d^{4} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right ) - \frac {2 \, {\left (2 \, {\left (8 \, {\left (\frac {2 \, {\left (b^{4} c^{3} d^{4} - 8 \, a b^{2} c^{4} d^{4} + 16 \, a^{2} c^{5} d^{4}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (b^{5} c^{2} d^{4} - 8 \, a b^{3} c^{3} d^{4} + 16 \, a^{2} b c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (3 \, b^{6} c d^{4} - 20 \, a b^{4} c^{2} d^{4} + 16 \, a^{2} b^{2} c^{3} d^{4} + 64 \, a^{3} c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{7} d^{4} + 4 \, a b^{5} c d^{4} - 80 \, a^{2} b^{3} c^{2} d^{4} + 192 \, a^{3} b c^{3} d^{4}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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