Optimal. Leaf size=515 \[ -\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.35, antiderivative size = 515, 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 = {746, 828, 857,
732, 435, 430} \begin {gather*} \frac {\sqrt {2} \sqrt {b^2-4 a c} \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 )}{5 c e^4 \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 {b^2-4 a c} \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 )}{5 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 746
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}-\frac {2 \int \frac {\frac {1}{2} c \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )-\frac {1}{2} c \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}{5 c e^3}\\ &=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}-\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}{5 e^4}+\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}{5 e^4}\\ &=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \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 )}{5 c e^4 \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} \sqrt {b^2-4 a c} (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 )}{5 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \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 )}{5 c e^4 \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} \sqrt {b^2-4 a c} (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 )}{5 c e^4 \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 = 29.84, size = 1111, normalized size = 2.16 \begin {gather*} \frac {4 e^2 (a+x (b+c x)) \left (e (7 b d-5 a e+2 b e x)+c \left (-8 d^2-2 d e x+e^2 x^2\right )\right )+\frac {4 e^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}}} \left (12 a^2 c e^2+\left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x (b+c x)+a \left (b^2 e^2+4 b c e (-4 d+3 e x)+4 c^2 \left (4 d^2+3 e^2 x^2\right )\right )\right )-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 ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 )-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 c e \left (a e^2-4 d \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 ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 )}{c \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}}}}}{10 e^5 \sqrt {d+e x} \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. \(4363\) vs.
\(2(453)=906\).
time = 1.00, size = 4364, normalized size = 8.47
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^{2}+b e x +a e \right ) \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 c x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{5 e^{2}}+\frac {2 \left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{3 c e}+\frac {2 \left (\frac {2 a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+2 d^{2} e b c -c^{2} d^{3}}{e^{4}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -c d \right )}{e^{4}}+\frac {b \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{3}}-\frac {2 d a c}{5 e^{2}}-\frac {2 \left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\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 {2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +c^{2} d^{2}}{e^{3}}+\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) c}{e^{3}}-\frac {2 c \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5 e^{2}}-\frac {2 \left (\frac {c \left (2 b e -c d \right )}{e^{2}}-\frac {2 c \left (2 b e +2 c d \right )}{5 e^{2}}\right ) \left (b e +c d \right )}{3 c e}\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}}\) | \(1177\) |
risch | \(\text {Expression too large to display}\) | \(1748\) |
default | \(\text {Expression too large to display}\) | \(4364\) |
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 = 0.50, size = 610, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left ({\left (16 \, c^{3} d^{4} + {\left (b^{3} - 12 \, a b c\right )} x e^{4} + {\left (6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d x + {\left (b^{3} - 12 \, a b c\right )} d\right )} e^{3} - 6 \, {\left (4 \, b c^{2} d^{2} x - {\left (b^{2} c + 4 \, a c^{2}\right )} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{3} d^{3} x - 3 \, b c^{2} 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 (16 \, c^{3} d^{3} e + {\left (b^{2} c + 12 \, a c^{2}\right )} x e^{4} - {\left (16 \, b c^{2} d x - {\left (b^{2} c + 12 \, a c^{2}\right )} d\right )} e^{3} + 16 \, {\left (c^{3} d^{2} x - b c^{2} d^{2}\right )} e^{2}\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 \, {\left (8 \, c^{3} d^{2} e^{2} - {\left (c^{3} x^{2} + 2 \, b c^{2} x - 5 \, a c^{2}\right )} e^{4} + {\left (2 \, c^{3} d x - 7 \, b c^{2} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{15 \, {\left (c^{2} x e^{6} + c^{2} d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \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 (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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