Optimal. Leaf size=208 \[ -\frac{2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac{2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}+\frac{2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}-\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2} \]
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Rubi [A] time = 0.215273, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ -\frac{2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac{2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}+\frac{2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}-\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx &=\int \left (-\frac{b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{d^2 f}+\frac{(-b c+a d)^2 (e+f x)^{5/2}}{d^2 (c+d x)}+\frac{b^2 (e+f x)^{7/2}}{d f}\right ) \, dx\\ &=-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}+\frac{(b c-a d)^2 \int \frac{(e+f x)^{5/2}}{c+d x} \, dx}{d^2}\\ &=\frac{2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}+\frac{\left ((b c-a d)^2 (d e-c f)\right ) \int \frac{(e+f x)^{3/2}}{c+d x} \, dx}{d^3}\\ &=\frac{2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac{2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}+\frac{\left ((b c-a d)^2 (d e-c f)^2\right ) \int \frac{\sqrt{e+f x}}{c+d x} \, dx}{d^4}\\ &=\frac{2 (b c-a d)^2 (d e-c f)^2 \sqrt{e+f x}}{d^5}+\frac{2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac{2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}+\frac{\left ((b c-a d)^2 (d e-c f)^3\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^5}\\ &=\frac{2 (b c-a d)^2 (d e-c f)^2 \sqrt{e+f x}}{d^5}+\frac{2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac{2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}+\frac{\left (2 (b c-a d)^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{d e}{f}+\frac{d x^2}{f}} \, dx,x,\sqrt{e+f x}\right )}{d^5 f}\\ &=\frac{2 (b c-a d)^2 (d e-c f)^2 \sqrt{e+f x}}{d^5}+\frac{2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac{2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2}-\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.454719, size = 175, normalized size = 0.84 \[ \frac{2 \left (\frac{105 (b c-a d)^2 (d e-c f) \left (\sqrt{d} \sqrt{e+f x} (-3 c f+4 d e+d f x)-3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )\right )}{d^{5/2}}-\frac{45 b d (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{f^2}+63 (e+f x)^{5/2} (b c-a d)^2+\frac{35 b^2 d^2 (e+f x)^{9/2}}{f^2}\right )}{315 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 972, normalized size = 4.7 \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 = 1.57661, size = 2234, normalized size = 10.74 \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 [A] time = 91.0127, size = 374, normalized size = 1.8 \begin{align*} \frac{2 b^{2} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{2}} + \frac{\left (e + f x\right )^{\frac{7}{2}} \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{7 d^{2} f^{2}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{5 d^{3}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{3 d^{4}} + \frac{\sqrt{e + f x} \left (2 a^{2} c^{2} d^{2} f^{2} - 4 a^{2} c d^{3} e f + 2 a^{2} d^{4} e^{2} - 4 a b c^{3} d f^{2} + 8 a b c^{2} d^{2} e f - 4 a b c d^{3} e^{2} + 2 b^{2} c^{4} f^{2} - 4 b^{2} c^{3} d e f + 2 b^{2} c^{2} d^{2} e^{2}\right )}{d^{5}} - \frac{2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{6} \sqrt{\frac{c f - d e}{d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.47214, size = 909, normalized size = 4.37 \begin{align*} -\frac{2 \,{\left (b^{2} c^{5} f^{3} - 2 \, a b c^{4} d f^{3} + a^{2} c^{3} d^{2} f^{3} - 3 \, b^{2} c^{4} d f^{2} e + 6 \, a b c^{3} d^{2} f^{2} e - 3 \, a^{2} c^{2} d^{3} f^{2} e + 3 \, b^{2} c^{3} d^{2} f e^{2} - 6 \, a b c^{2} d^{3} f e^{2} + 3 \, a^{2} c d^{4} f e^{2} - b^{2} c^{2} d^{3} e^{3} + 2 \, a b c d^{4} e^{3} - a^{2} d^{5} e^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{5}} + \frac{2 \,{\left (35 \,{\left (f x + e\right )}^{\frac{9}{2}} b^{2} d^{8} f^{16} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{2} c d^{7} f^{17} + 90 \,{\left (f x + e\right )}^{\frac{7}{2}} a b d^{8} f^{17} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} c^{2} d^{6} f^{18} - 126 \,{\left (f x + e\right )}^{\frac{5}{2}} a b c d^{7} f^{18} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} a^{2} d^{8} f^{18} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c^{3} d^{5} f^{19} + 210 \,{\left (f x + e\right )}^{\frac{3}{2}} a b c^{2} d^{6} f^{19} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} c d^{7} f^{19} + 315 \, \sqrt{f x + e} b^{2} c^{4} d^{4} f^{20} - 630 \, \sqrt{f x + e} a b c^{3} d^{5} f^{20} + 315 \, \sqrt{f x + e} a^{2} c^{2} d^{6} f^{20} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{2} d^{8} f^{16} e + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c^{2} d^{6} f^{18} e - 210 \,{\left (f x + e\right )}^{\frac{3}{2}} a b c d^{7} f^{18} e + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} d^{8} f^{18} e - 630 \, \sqrt{f x + e} b^{2} c^{3} d^{5} f^{19} e + 1260 \, \sqrt{f x + e} a b c^{2} d^{6} f^{19} e - 630 \, \sqrt{f x + e} a^{2} c d^{7} f^{19} e + 315 \, \sqrt{f x + e} b^{2} c^{2} d^{6} f^{18} e^{2} - 630 \, \sqrt{f x + e} a b c d^{7} f^{18} e^{2} + 315 \, \sqrt{f x + e} a^{2} d^{8} f^{18} e^{2}\right )}}{315 \, d^{9} f^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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