Optimal. Leaf size=641 \[ -\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.58, antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 846, 857,
732, 435, 430} \begin {gather*} -\frac {4 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 752
Rule 846
Rule 857
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {(d+e x)^{3/2} \left (-\frac {5}{2} e (b d-2 a e)-\frac {5}{2} e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac {4 \int \frac {\sqrt {d+e x} \left (-\frac {5}{4} e \left (b^2 d e-16 a c d e+3 b \left (c d^2+a e^2\right )\right )-\frac {5}{2} e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac {8 \int \frac {\frac {5}{8} e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )-\frac {5}{8} e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (8 \left (\frac {5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac {5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right ) e}\\ &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left (\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (16 \sqrt {2} \left (\frac {5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac {5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 c^3 \sqrt {b^2-4 a c} e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 30.95, size = 890, normalized size = 1.39 \begin {gather*} \frac {\frac {4 e (-2 c d+b e) \left (-3 c^2 d^2-8 b^2 e^2+c e (3 b d+29 a e)\right ) (a+x (b+c x))}{\sqrt {d+e x}}-4 c \sqrt {d+e x} \left (-4 b^3 e^3 x-b^2 e^2 (4 a e+c x (-9 d+e x))+b c \left (3 c d^2 (d-3 e x)+a e^2 (9 d+13 e x)\right )+2 c \left (5 a^2 e^3+3 c^2 d^3 x+a c e \left (-9 d^2-9 d e x+2 e^2 x^2\right )\right )\right )+\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left ((-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (8 b^4 e^4-b^3 e^3 \left (27 c d+8 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+b^2 c e^2 \left (27 c d^2-37 a e^2+19 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+2 c^2 \left (10 a^2 e^4+3 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}-a d e^2 \left (54 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+b c e \left (-9 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (108 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{6 c^3 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6486\) vs.
\(2(575)=1150\).
time = 0.91, size = 6487, normalized size = 10.12
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (c e x +c d \right ) \left (-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a -b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) x}{c^{3} \left (4 a c -b^{2}\right )}-\frac {2 a^{2} c \,e^{3}-b^{2} e^{3} a +3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +d^{3} b \,c^{2}}{c^{3} \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (c e x +c d \right )}}+\frac {2 e^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{3 c^{2}}+\frac {2 \left (-\frac {e^{2} \left (a c \,e^{2}-b^{2} e^{2}+4 b c d e -6 c^{2} d^{2}\right )}{c^{3}}+\frac {4 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+18 a b \,c^{2} d \,e^{3}-24 d^{2} e^{2} c^{3} a +b^{4} e^{4}-5 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}-6 b \,c^{3} d^{3} e +4 c^{4} d^{4}}{c^{3} \left (4 a c -b^{2}\right )}-\frac {e \left (2 a^{2} c \,e^{3}-b^{2} e^{3} a +3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +d^{3} b \,c^{2}\right )}{c^{2} \left (4 a c -b^{2}\right )}-\frac {2 d \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a -b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right )}{c^{2} \left (4 a c -b^{2}\right )}-\frac {2 e^{3} \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c^{2}}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}+\frac {2 \left (-\frac {e^{3} \left (b e -4 c d \right )}{c^{2}}-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a -b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) e}{c^{2} \left (4 a c -b^{2}\right )}-\frac {2 e^{3} \left (b e +c d \right )}{3 c^{2}}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\) | \(1364\) |
risch | \(\text {Expression too large to display}\) | \(2856\) |
default | \(\text {Expression too large to display}\) | \(6487\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.72, size = 1013, normalized size = 1.58 \begin {gather*} -\frac {2 \, {\left ({\left (6 \, c^{5} d^{4} x^{2} + 6 \, b c^{4} d^{4} x + 6 \, a c^{4} d^{4} - {\left (8 \, a b^{4} - 41 \, a^{2} b^{2} c + 30 \, a^{3} c^{2} + {\left (8 \, b^{4} c - 41 \, a b^{2} c^{2} + 30 \, a^{2} c^{3}\right )} x^{2} + {\left (8 \, b^{5} - 41 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} x\right )} e^{4} + {\left ({\left (23 \, b^{3} c^{2} - 104 \, a b c^{3}\right )} d x^{2} + {\left (23 \, b^{4} c - 104 \, a b^{2} c^{2}\right )} d x + {\left (23 \, a b^{3} c - 104 \, a^{2} b c^{2}\right )} d\right )} e^{3} - {\left ({\left (17 \, b^{2} c^{3} - 104 \, a c^{4}\right )} d^{2} x^{2} + {\left (17 \, b^{3} c^{2} - 104 \, a b c^{3}\right )} d^{2} x + {\left (17 \, a b^{2} c^{2} - 104 \, a^{2} c^{3}\right )} d^{2}\right )} e^{2} - 12 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) - 3 \, {\left ({\left (8 \, a b^{3} c - 29 \, a^{2} b c^{2} + {\left (8 \, b^{3} c^{2} - 29 \, a b c^{3}\right )} x^{2} + {\left (8 \, b^{4} c - 29 \, a b^{2} c^{2}\right )} x\right )} e^{4} - {\left ({\left (19 \, b^{2} c^{3} - 58 \, a c^{4}\right )} d x^{2} + {\left (19 \, b^{3} c^{2} - 58 \, a b c^{3}\right )} d x + {\left (19 \, a b^{2} c^{2} - 58 \, a^{2} c^{3}\right )} d\right )} e^{3} + 9 \, {\left (b c^{4} d^{2} x^{2} + b^{2} c^{3} d^{2} x + a b c^{3} d^{2}\right )} e^{2} - 6 \, {\left (c^{5} d^{3} x^{2} + b c^{4} d^{3} x + a c^{4} d^{3}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) - 3 \, \sqrt {c x^{2} + b x + a} {\left ({\left (4 \, a b^{2} c^{2} - 10 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (4 \, b^{3} c^{2} - 13 \, a b c^{3}\right )} x\right )} e^{4} - 9 \, {\left (a b c^{3} d + {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d x\right )} e^{3} + 9 \, {\left (b c^{4} d^{2} x + 2 \, a c^{4} d^{2}\right )} e^{2} - 3 \, {\left (2 \, c^{5} d^{3} x + b c^{4} d^{3}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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