Optimal. Leaf size=540 \[ -\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {7 \sqrt {2} \sqrt {b^2-4 a c} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {56 \sqrt {2} \sqrt {b^2-4 a c} e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.44, antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {782, 756, 846,
857, 732, 435, 430} \begin {gather*} -\frac {56 \sqrt {2} e \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 )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {7 \sqrt {2} e \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 (-c e (9 a e+23 b d)+8 b^2 e^2+23 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 )}{15 c^3 \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 {56 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 756
Rule 782
Rule 846
Rule 857
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+(7 e) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {(14 e) \int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (5 c d^2-e (b d+3 a e)\right )+2 e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {(28 e) \int \frac {\frac {1}{4} \left (15 c^2 d^3+4 b e^2 (b d+a e)-c d e (11 b d+17 a e)\right )+\frac {1}{4} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2}\\ &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {\left (28 e (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}{15 c^2}+\frac {\left (7 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2}\\ &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (7 \sqrt {2} \sqrt {b^2-4 a c} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 )}{15 c^3 \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 (56 \sqrt {2} \sqrt {b^2-4 a c} e (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 )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {7 \sqrt {2} \sqrt {b^2-4 a c} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 )}{15 c^3 \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 {56 \sqrt {2} \sqrt {b^2-4 a c} e (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 )}{15 c^3 \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 = 27.41, size = 780, normalized size = 1.44 \begin {gather*} \frac {2 \sqrt {d+e x} \left (c \left (-28 b e^3 (a+b x)+c^2 \left (-15 d^3-45 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+7 c e^2 (b x (11 d-e x)+a (11 d+3 e x))\right )+7 (d+e x) \left (\frac {e^2 \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) (a+x (b+c x))}{(d+e x)^2}-\frac {i \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 {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)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 (-30 c^3 d^3+8 b^2 e^2 \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )-c^2 d \left (-45 b d e-34 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 b^2 d e-17 a b e^2+23 b d \sqrt {\left (b^2-4 a c\right ) e^2}+9 a e \sqrt {\left (b^2-4 a c\right ) e^2}\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 )}{2 \sqrt {2} \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}}} \sqrt {d+e x}}\right )\right )}{15 c^3 \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. \(4801\) vs.
\(2(472)=944\).
time = 1.24, size = 4802, normalized size = 8.89
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 (a c \,e^{2}-b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e x}{c^{3}}+\frac {a b \,e^{3}-3 a d \,e^{2} c +c^{2} d^{3}}{c^{3}}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (c e x +c d \right )}}+\frac {4 e^{3} x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{5 c}+\frac {2 \left (-\frac {e^{3} \left (b e -8 c d \right )}{c}-\frac {4 e^{3} \left (2 b e +2 c d \right )}{5 c}\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 {e \left (3 c \,e^{3} a b -8 d \,e^{2} c^{2} a -b^{3} e^{3}+4 b^{2} d \,e^{2} c -6 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right )}{c^{3}}-\frac {\left (3 c \,e^{3} a b -8 d \,e^{2} c^{2} a -b^{3} e^{3}+5 b^{2} d \,e^{2} c -9 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right ) e}{c^{3}}+\frac {e \left (a b \,e^{3}-3 a d \,e^{2} c +c^{2} d^{3}\right )}{c^{2}}-\frac {2 d \left (a c \,e^{2}-b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e}{c^{2}}-\frac {4 a d \,e^{3}}{5 c}-\frac {2 \left (-\frac {e^{3} \left (b e -8 c d \right )}{c}-\frac {4 e^{3} \left (2 b e +2 c d \right )}{5 c}\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 {e^{2} \left (2 a c \,e^{2}-b^{2} e^{2}+4 b c d e -12 c^{2} d^{2}\right )}{c^{2}}-\frac {\left (a c \,e^{2}-b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e^{2}}{c^{2}}-\frac {4 e^{3} \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5 c}-\frac {2 \left (-\frac {e^{3} \left (b e -8 c d \right )}{c}-\frac {4 e^{3} \left (2 b e +2 c d \right )}{5 c}\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}}\) | \(1346\) |
risch | \(\text {Expression too large to display}\) | \(1834\) |
default | \(\text {Expression too large to display}\) | \(4802\) |
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 = 0.47, size = 767, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left (7 \, {\left (22 \, c^{4} d^{3} x^{2} + 22 \, b c^{3} d^{3} x + 22 \, a c^{3} d^{3} - {\left (8 \, a b^{3} - 21 \, a^{2} b c + {\left (8 \, b^{3} c - 21 \, a b c^{2}\right )} x^{2} + {\left (8 \, b^{4} - 21 \, a b^{2} c\right )} x\right )} e^{3} + 3 \, {\left ({\left (9 \, b^{2} c^{2} - 14 \, a c^{3}\right )} d x^{2} + {\left (9 \, b^{3} c - 14 \, a b c^{2}\right )} d x + {\left (9 \, a b^{2} c - 14 \, a^{2} c^{2}\right )} d\right )} e^{2} - 33 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x + a b c^{2} d^{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 ) - 21 \, {\left ({\left (8 \, a b^{2} c - 9 \, a^{2} c^{2} + {\left (8 \, b^{2} c^{2} - 9 \, a c^{3}\right )} x^{2} + {\left (8 \, b^{3} c - 9 \, a b c^{2}\right )} x\right )} e^{3} - 23 \, {\left (b c^{3} d x^{2} + b^{2} c^{2} d x + a b c^{2} d\right )} e^{2} + 23 \, {\left (c^{4} d^{2} x^{2} + b c^{3} d^{2} x + a c^{3} d^{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 \, {\left (45 \, c^{4} d^{2} x e + 15 \, c^{4} d^{3} - {\left (6 \, c^{4} x^{3} - 7 \, b c^{3} x^{2} - 28 \, a b c^{2} - 7 \, {\left (4 \, b^{2} c^{2} - 3 \, a c^{3}\right )} x\right )} e^{3} - {\left (32 \, c^{4} d x^{2} + 77 \, b c^{3} d x + 77 \, a c^{3} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{5} x^{2} + b c^{4} x + a c^{4}\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 (b+2\,c\,x\right )\,{\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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