3.6.0 ∫ x(d+ex)3∕2 ________________

Integrand size = 23, antiderivative size = 453 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 (c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c}+\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )+b c \left (c d^2+e \left (2 \sqrt {b^2-4 a c} d-3 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b^3 e^2-b^2 e \left (2 c d-\sqrt {b^2-4 a c} e\right )+b c \left (c d^2-e \left (2 \sqrt {b^2-4 a c} d+3 a e\right )\right )-c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

Rubi [A] (veriﬁed)

Time = 2.90 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {838, 840, 1180, 214} \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \left (b c \left (e \left (2 d \sqrt {b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )\right )-b^2 e \left (e \sqrt {b^2-4 a c}+2 c d\right )+b^3 e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (b c \left (c d^2-e \left (2 d \sqrt {b^2-4 a c}+3 a e\right )\right )-c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )\right )-b^2 e \left (2 c d-e \sqrt {b^2-4 a c}\right )+b^3 e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x} (c d-b e)}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c} \]

Rubi steps \begin{align*} \text {integral}& = \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*}

Mathematica [C] (veriﬁed)

Time = 1.75 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.09 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {d+e x} (4 c d-3 b e+c e x)+\frac {3 \left (i b^3 e^2+b^2 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-3 a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (-i b^3 e^2+b^2 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+b c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+3 i a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{3 c^{5/2}} \]

Maple [A] (veriﬁed)

Time = 0.70 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.04


method	result	size

pseudoelliptic	\(-\frac {-\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (a c -b^{2}\right ) e^{2}+2 b c d e -c^{2} d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-3 a b c +b^{3}\right ) e^{3}+2 d \left (2 a \,c^{2}-b^{2} c \right ) e^{2}+b \,c^{2} d^{2} e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (\left (\left (a c -b^{2}\right ) e^{2}+2 b c d e -c^{2} d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (3 a b c -b^{3}\right ) e^{3}+2 d \left (-2 a \,c^{2}+b^{2} c \right ) e^{2}-b \,c^{2} d^{2} e \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+2 \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \left (\left (-\frac {c x}{3}+b \right ) e -\frac {4 c d}{3}\right )\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c^{2}}\)	\(473\)
risch	\(-\frac {2 \left (-x c e +3 b e -4 c d \right ) \sqrt {e x +d}}{3 c^{2}}-\frac {8 \left (-\frac {\left (-3 c \,e^{3} b a +4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-2 b^{2} c d \,e^{2}+b \,c^{2} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (3 c \,e^{3} b a -4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+2 b^{2} c d \,e^{2}-b \,c^{2} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\)	\(512\)
derivativedivides	\(-\frac {2 \left (-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 c \,e^{3} b a +4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-2 b^{2} c d \,e^{2}+b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 c \,e^{3} b a -4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+2 b^{2} c d \,e^{2}-b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)
default	\(-\frac {2 \left (-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-c d \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-3 c \,e^{3} b a +4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-2 b^{2} c d \,e^{2}+b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 c \,e^{3} b a -4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+2 b^{2} c d \,e^{2}-b \,c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(521\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5572 vs. \(2 (392) = 784\).

Time = 1.65 (sec) , antiderivative size = 5572, normalized size of antiderivative = 12.30 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

Sympy [F(-1)]

Timed out. \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Timed out} \]

Maxima [F]

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

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (392) = 784\).

Time = 0.36 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.18 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} + 3 \, \sqrt {e x + d} c^{2} d - 3 \, \sqrt {e x + d} b c e\right )}}{3 \, c^{3}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2}\right )} c^{2} e^{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 | e \right |} + {\left (2 \, b c^{5} d^{3} e - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e^{2} + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{3} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{4}\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 {e x + 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} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2}\right )} c^{2} e^{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 | e \right |} + {\left (2 \, b c^{5} d^{3} e - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e^{2} + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{3} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{4}\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 {e x + 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} {\left | e \right |}} \]

Mupad [B] (veriﬁcation not implemented)

Time = 13.80 (sec) , antiderivative size = 13841, normalized size of antiderivative = 30.55 \[ \int \frac {x (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

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

Optimal result