∫ x(d+ex)3∕2 ________________

Optimal. Leaf size=453 \[ \frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}+\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )+b c \left (c d^2+e \left (2 \sqrt {b^2-4 a c} d-3 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d-\sqrt {b^2-4 a c} e\right )+b c \left (c d^2-e \left (2 \sqrt {b^2-4 a c} d+3 a e\right )\right )-c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

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Rubi [A]
time = 2.94, antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {838, 840, 1180, 214} \begin {gather*} \frac {\sqrt {2} \left (b c \left (e \left (2 d \sqrt {b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )\right )-b^2 e \left (e \sqrt {b^2-4 a c}+2 c d\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (b c \left (c d^2-e \left (2 d \sqrt {b^2-4 a c}+3 a e\right )\right )-c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )\right )-b^2 e \left (2 c d-e \sqrt {b^2-4 a c}\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x} (c d-b e)}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c} \end {gather*}

\begin {align*} \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx &=\frac {2 (d+e x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {d+e x} (-a e+(c d-b e) x)}{a+b x+c x^2} \, dx}{c}\\ &=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}+\frac {\int \frac {-a e (2 c d-b e)+\left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c^2}\\ &=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}+\frac {2 \text {Subst}\left (\int \frac {-a e^2 (2 c d-b e)-d \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )+\left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2}\\ &=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}-\frac {\left (b^3 e^2-b^2 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )+b c \left (c d^2+e \left (2 \sqrt {b^2-4 a c} d-3 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {\left (b^3 e^2-b^2 e \left (2 c d-\sqrt {b^2-4 a c} e\right )+b c \left (c d^2-e \left (2 \sqrt {b^2-4 a c} d+3 a e\right )\right )-c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{c^2 \sqrt {b^2-4 a c}}\\ &=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}+\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )+b c \left (c d^2+e \left (2 \sqrt {b^2-4 a c} d-3 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d-\sqrt {b^2-4 a c} e\right )+b c \left (c d^2-e \left (2 \sqrt {b^2-4 a c} d+3 a e\right )\right )-c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.73, size = 493, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {c} \sqrt {d+e x} (4 c d-3 b e+c e x)+\frac {3 \left (i b^3 e^2+b^2 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-3 a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (-i b^3 e^2+b^2 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+b c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+3 i a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{3 c^{5/2}} \end {gather*}

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method	result	size

derivativedivides	\(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+b e \sqrt {e x +d}-c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-2 b^{2} c d \,e^{2}+b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+2 b^{2} c d \,e^{2}-b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)
default	\(-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+b e \sqrt {e x +d}-c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-2 b^{2} c d \,e^{2}+b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+2 b^{2} c d \,e^{2}-b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)
risch	\(\text {Expression too large to display}\)	\(1701\)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5507 vs. \(2 (406) = 812\).
time = 3.90, size = 5507, normalized size = 12.16 \begin {gather*} \text {Too large to display} \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (406) = 812\).
time = 2.15, size = 978, normalized size = 2.16 \begin {gather*} \frac {{\left ({\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{3} - 2 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{3} + {\left (b^{2} c^{2} + a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, b c^{5} d^{3} - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{2} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e + \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{5} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{4} d e + \sqrt {b^{2} - 4 \, a c} a c^{4} e^{2}\right )} c^{2}} - \frac {{\left ({\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{3} - 2 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{3} + {\left (b^{2} c^{2} + a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, b c^{5} d^{3} - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{2} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e - \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{5} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{4} d e + \sqrt {b^{2} - 4 \, a c} a c^{4} e^{2}\right )} c^{2}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} + 3 \, \sqrt {x e + d} c^{2} d - 3 \, \sqrt {x e + d} b c e\right )}}{3 \, c^{3}} \end {gather*}

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Mupad [B]
time = 4.72, size = 2500, normalized size = 5.52 \begin {gather*} \text {Too large to display} \end {gather*}

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3.6.36 \(\int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [536]