3.23.0 ∫ (d+ex)5∕2 _______________

Integrand size = 22, antiderivative size = 459 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\sqrt {2} \left (2 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d-\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d-3 b \sqrt {b^2-4 a c} d+3 a b e-a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt {b^2-4 a c} e+3 b \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

Rubi [A] (veriﬁed)

Time = 3.11 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {717, 838, 840, 1180, 214} \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=-\frac {\sqrt {2} \left (-3 c^2 d e \left (-d \sqrt {b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+2 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (-3 c^2 d e \left (d \sqrt {b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt {b^2-4 a c}+a e\right )+a e \sqrt {b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+2 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c} \]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {d+e x} \left (c d^2-a e^2+e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{c} \\ & = \frac {2 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c^2} \\ & = \frac {2 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {2 \text {Subst}\left (\int \frac {e \left (c^2 d^3-3 a c d e^2+a b e^3\right )-d e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )+e \left (3 c^2 d^2+b^2 e^2-c e (3 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 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\left (2 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d-\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d-3 b \sqrt {b^2-4 a c} d+3 a b e-a \sqrt {b^2-4 a c} e\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 (2 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt {b^2-4 a c} e+3 b \left (\sqrt {b^2-4 a c} d+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 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\sqrt {2} \left (2 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d-\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d-3 b \sqrt {b^2-4 a c} d+3 a b e-a \sqrt {b^2-4 a c} e\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 (2 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt {b^2-4 a c} e+3 b \left (\sqrt {b^2-4 a c} d+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 = 1.77 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x} (7 c d-3 b e+c e x)+\frac {3 \left (-2 i c^3 d^3+b^2 \left (i b+\sqrt {-b^2+4 a c}\right ) e^3+3 c^2 d e \left (i b d+\sqrt {-b^2+4 a c} d+2 i a e\right )-c e^2 \left (3 i b^2 d+3 b \sqrt {-b^2+4 a c} d+3 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (2 i c^3 d^3+b^2 \left (-i b+\sqrt {-b^2+4 a c}\right ) e^3+3 c^2 d e \left (-i b d+\sqrt {-b^2+4 a c} d-2 i a e\right )+c e^2 \left (3 i b^2 d-3 b \sqrt {-b^2+4 a c} d+3 i a b e-a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{3 c^{5/2}} \]

Maple [A] (veriﬁed)

Time = 0.69 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.03


method	result	size

pseudoelliptic	\(-\frac {\left (-\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right )\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\left (\sqrt {2}\, \left (\left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right )\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+2 \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \left (\frac {\left (-e x -7 d \right ) c}{3}+b e \right ) \sqrt {e x +d}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right ) e}{\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, c^{2}}\)	\(472\)
risch	\(-\frac {2 e \left (-c e x +3 b e -7 c d \right ) \sqrt {e x +d}}{3 c^{2}}-\frac {8 e \left (-\frac {\left (-3 a b c \,e^{3}+6 a \,c^{2} d \,e^{2}+b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -2 c^{3} d^{3}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\)	\(531\)
derivativedivides	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-2 c d \sqrt {e x +d}}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+6 a \,c^{2} d \,e^{2}+b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -2 c^{3} d^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\right )\)	\(543\)
default	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-2 c d \sqrt {e x +d}}{c^{2}}+\frac {\frac {\left (-3 a b c \,e^{3}+6 a \,c^{2} d \,e^{2}+b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -2 c^{3} d^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\right )\)	\(543\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6701 vs. \(2 (397) = 794\).

Time = 3.24 (sec) , antiderivative size = 6701, normalized size of antiderivative = 14.60 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Timed out} \]

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} + b x + a} \,d x } \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (397) = 794\).

Time = 0.36 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.24 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e - 3 \, \sqrt {e x + d} b c e^{2}\right )}}{3 \, c^{3}} + \frac {{\left ({\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\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} e^{2} - 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{3} e - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{3}\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 (4 \, c^{6} d^{4} e - 8 \, b c^{5} d^{3} e^{2} + 3 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} e^{3} - {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d e^{4} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{5}\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 ({\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\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} e^{2} + 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{3} e - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{3}\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 (4 \, c^{6} d^{4} e - 8 \, b c^{5} d^{3} e^{2} + 3 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} e^{3} - {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d e^{4} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{5}\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 = 12.59 (sec) , antiderivative size = 16475, normalized size of antiderivative = 35.89 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

\(\int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx\) [2289]

Optimal result