Optimal. Leaf size=207 \[ -\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]
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Rubi [A] time = 0.0885637, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 621, 206} \[ -\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac{(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{2 c}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac{\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac{\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \sqrt{a+b x+c x^2} \, dx}{256 c^3}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^4}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^4}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{1024 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^4}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.278456, size = 180, normalized size = 0.87 \[ \frac{(2 c d-b e) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{6144 c^{9/2}}+\frac{e (a+x (b+c x))^{7/2}}{7 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 807, normalized size = 3.9 \begin{align*} \text{result too large to display} \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 [B] time = 3.28142, size = 1974, normalized size = 9.54 \begin{align*} \text{result too large to display} \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 \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15808, size = 601, normalized size = 2.9 \begin{align*} \frac{1}{21504} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, c^{2} x e + \frac{14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac{70 \, b c^{7} d + 37 \, b^{2} c^{6} e + 72 \, a c^{7} e}{c^{6}}\right )} x + \frac{378 \, b^{2} c^{6} d + 728 \, a c^{7} d + 3 \, b^{3} c^{5} e + 788 \, a b c^{6} e}{c^{6}}\right )} x + \frac{14 \, b^{3} c^{5} d + 2184 \, a b c^{6} d - 7 \, b^{4} c^{4} e + 60 \, a b^{2} c^{5} e + 1152 \, a^{2} c^{6} e}{c^{6}}\right )} x - \frac{70 \, b^{4} c^{4} d - 672 \, a b^{2} c^{5} d - 7392 \, a^{2} c^{6} d - 35 \, b^{5} c^{3} e + 336 \, a b^{3} c^{4} e - 912 \, a^{2} b c^{5} e}{c^{6}}\right )} x + \frac{210 \, b^{5} c^{3} d - 2240 \, a b^{3} c^{4} d + 7392 \, a^{2} b c^{5} d - 105 \, b^{6} c^{2} e + 1120 \, a b^{4} c^{3} e - 3696 \, a^{2} b^{2} c^{4} e + 3072 \, a^{3} c^{5} e}{c^{6}}\right )} + \frac{5 \,{\left (2 \, b^{6} c d - 24 \, a b^{4} c^{2} d + 96 \, a^{2} b^{2} c^{3} d - 128 \, a^{3} c^{4} d - b^{7} e + 12 \, a b^{5} c e - 48 \, a^{2} b^{3} c^{2} e + 64 \, a^{3} b c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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