Optimal. Leaf size=590 \[ -\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {16 \sqrt {2} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.40, antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 834, 857,
732, 435, 430} \begin {gather*} -\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 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 \left (b^2-4 a c\right )^{3/2} \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 {16 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c 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 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \left (-x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-8 b \left (a e^2+c d^2\right )+4 a c d e+7 b^2 d e\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 752
Rule 834
Rule 857
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (8 c d^2-7 b d e+6 a e^2\right )+\frac {1}{2} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} e \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )-\frac {1}{4} e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {\left (8 (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}-\frac {\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 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 \left (b^2-4 a c\right )^{3/2} \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} (2 c d-b e) \left (c d^2-b d e+a e^2\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 )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 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 \left (b^2-4 a c\right )^{3/2} \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 {16 \sqrt {2} (2 c d-b e) \left (c d^2-b d e+a 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 \left (b^2-4 a c\right )^{3/2} \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 = 1460, normalized size = 2.47 \begin {gather*} \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^3 \left (\frac {2 \left (b c d^2-4 a c d e+a b e^2+2 c^2 d^2 x-2 b c d e x+b^2 e^2 x-2 a c e^2 x\right )}{3 c \left (-b^2+4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {2 \left (8 b c^2 d^2-9 b^2 c d e+4 a c^2 d e+b^3 e^2+4 a b c e^2+16 c^3 d^2 x-16 b c^2 d e x+b^2 c e^2 x+12 a c^2 e^2 x\right )}{3 c \left (-b^2+4 a c\right )^2 \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{5/2}}+\frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \left (-4 \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}}} \left (16 c^3 d^2 \left (-1+\frac {d}{d+e x}\right )^2+\frac {b^2 e^3 \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}+4 c^2 e \left (a e \left (3+\frac {7 d^2}{(d+e x)^2}-\frac {6 d}{d+e x}\right )-4 b d \left (1+\frac {2 d^2}{(d+e x)^2}-\frac {3 d}{d+e x}\right )\right )+c e^2 \left (\frac {12 a^2 e^2}{(d+e x)^2}+b^2 \left (1+\frac {17 d^2}{(d+e x)^2}-\frac {18 d}{d+e x}\right )-\frac {4 a b e \left (-3+\frac {7 d}{d+e x}\right )}{d+e x}\right )\right )+\frac {i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} 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 )}{\sqrt {d+e x}}-\frac {i \sqrt {2} \left (-b^3 e^3+b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-2 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} 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 )}{\sqrt {d+e x}}\right )}{6 c \left (-b^2+4 a c\right )^2 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}}} (a+x (b+c x))^{5/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \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. \(12989\) vs.
\(2(526)=1052\).
time = 0.86, size = 12990, normalized size = 22.02
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\left (-\frac {2 \left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) x}{3 c^{3} \left (4 a c -b^{2}\right )}+\frac {\frac {2}{3} a b \,e^{2}-\frac {8}{3} a c d e +\frac {2}{3} b c \,d^{2}}{c^{3} \left (4 a c -b^{2}\right )}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {\left (12 a c \,e^{2}+b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right ) x}{3 c \left (4 a c -b^{2}\right )^{2}}-\frac {4 a b c \,e^{2}+4 a \,c^{2} d e +b^{3} e^{2}-9 b^{2} d c e +8 d^{2} b \,c^{2}}{3 \left (4 a c -b^{2}\right )^{2} c^{2}}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (c e x +c d \right )}}+\frac {2 \left (-\frac {4 a b c \,e^{3}-32 d \,e^{2} c^{2} a -b^{3} e^{3}+32 b \,c^{2} d^{2} e -32 c^{3} d^{3}}{3 \left (4 a c -b^{2}\right )^{2} c}-\frac {e \left (4 a b c \,e^{2}+4 a \,c^{2} d e +b^{3} e^{2}-9 b^{2} d c e +8 d^{2} b \,c^{2}\right )}{3 c \left (4 a c -b^{2}\right )^{2}}-\frac {2 d \left (12 a c \,e^{2}+b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right )}{3 \left (4 a c -b^{2}\right )^{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 e \left (12 a c \,e^{2}+b^{2} e^{2}-16 b c d e +16 c^{2} d^{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 )}{3 \left (4 a c -b^{2}\right )^{2} \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}}\) | \(1229\) |
default | \(\text {Expression too large to display}\) | \(12990\) |
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 = 0.55, size = 1317, normalized size = 2.23 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{5} d^{3} x^{4} + 32 \, b c^{4} d^{3} x^{3} + 32 \, a b c^{3} d^{3} x + 16 \, a^{2} c^{3} d^{3} + 16 \, {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{3} x^{2} + {\left (a^{2} b^{3} - 12 \, a^{3} b c + {\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{4} + 2 \, {\left (b^{4} c - 12 \, a b^{2} c^{2}\right )} x^{3} + {\left (b^{5} - 10 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left (a b^{4} - 12 \, a^{2} b^{2} c\right )} x\right )} e^{3} + 6 \, {\left ({\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d x^{4} + 2 \, {\left (b^{3} c^{2} + 4 \, a b c^{3}\right )} d x^{3} + {\left (b^{4} c + 6 \, a b^{2} c^{2} + 8 \, a^{2} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{3} c + 4 \, a^{2} b c^{2}\right )} d x + {\left (a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d\right )} e^{2} - 24 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} d^{2} x + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{2} x^{2}\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 (a^{2} b^{2} c + 12 \, a^{3} c^{2} + {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{4} + 2 \, {\left (b^{3} c^{2} + 12 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c + 14 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c + 12 \, a^{2} b c^{2}\right )} x\right )} e^{3} - 16 \, {\left (b c^{4} d x^{4} + 2 \, b^{2} c^{3} d x^{3} + 2 \, a b^{2} c^{2} d x + a^{2} b c^{2} d + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d x^{2}\right )} e^{2} + 16 \, {\left (c^{5} d^{2} x^{4} + 2 \, b c^{4} d^{2} x^{3} + 2 \, a b c^{3} d^{2} x + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{2} x^{2}\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 (8 \, a^{2} b c^{2} + {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{3} + 2 \, {\left (b^{3} c^{2} + 8 \, a b c^{3}\right )} x^{2} + {\left (11 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x\right )} e^{3} - {\left (16 \, b c^{4} d x^{3} + {\left (25 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{2} + {\left (7 \, b^{3} c^{2} + 20 \, a b c^{3}\right )} d x + {\left (5 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d\right )} e^{2} + {\left (16 \, c^{5} d^{2} x^{3} + 24 \, b c^{4} d^{2} x^{2} + 6 \, {\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d^{2} x - {\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{3} + {\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\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 )}^{5/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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