Optimal. Leaf size=617 \[ -\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 617, 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 = {746, 848, 857,
732, 435, 430} \begin {gather*} -\frac {2 \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 (-c e (3 a e+b d)+b^2 e^2+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 e^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \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) \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 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{15 e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e)}{15 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 746
Rule 848
Rule 857
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {\int \frac {b+2 c x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{5 e}\\ &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (-b c d+2 b^2 e-6 a c e\right )-\frac {1}{2} c (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{15 e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {1}{4} c \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )-\frac {1}{2} c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {(c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^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 (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^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 {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 e^2 \left (c d^2-b d e+a e^2\right ) \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 = 30.67, size = 1368, normalized size = 2.22 \begin {gather*} \sqrt {d+e x} \sqrt {a+x (b+c x)} \left (-\frac {2}{5 e (d+e x)^3}-\frac {2 (-2 c d+b e)}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {4 \left (-c^2 d^2+b c d e-b^2 e^2+3 a c e^2\right )}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}\right )-\frac {(d+e x)^{3/2} \sqrt {a+x (b+c x)} \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 (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}-c^2 e \left (a e \left (3+\frac {2 d^2}{(d+e x)^2}-\frac {6 d}{d+e x}\right )+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 {3 a^2 e^2}{(d+e x)^2}+b^2 \left (1+\frac {2 d^2}{(d+e x)^2}-\frac {3 d}{d+e x}\right )+\frac {a b e \left (-3+\frac {2 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 (c^2 d^2+b^2 e^2-c e (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 )+b c e \left (4 a e^2-d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-a e^2 \left (8 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 )}{15 e^3 \left (c d^2-b d e+a e^2\right )^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 {a+b x+c x^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. \(12979\) vs.
\(2(547)=1094\).
time = 1.06, size = 12980, normalized size = 21.04
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x b d +a d}}{15 e^{3} \left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 \left (c e \,x^{2}+b e x +a e \right ) \left (3 a c \,e^{2}-b^{2} e^{2}+b c d e -c^{2} d^{2}\right )}{15 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} e^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 \left (-\frac {c \left (b e -2 c d \right )}{15 e^{2} \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {2 \left (b e -c d \right ) \left (3 a c \,e^{2}-b^{2} e^{2}+b c d e -c^{2} d^{2}\right )}{15 e^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {2 b \left (3 a c \,e^{2}-b^{2} e^{2}+b c d e -c^{2} d^{2}\right )}{15 e \left (e^{2} a -b d e +c \,d^{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 {4 c \left (3 a c \,e^{2}-b^{2} e^{2}+b c d e -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 )}{15 e \left (e^{2} a -b d e +c \,d^{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}}\) | \(1149\) |
default | \(\text {Expression too large to display}\) | \(12980\) |
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 = 1.16, size = 1162, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left ({\left (2 \, c^{3} d^{6} + {\left (2 \, b^{3} - 9 \, a b c\right )} x^{3} e^{6} - 3 \, {\left ({\left (b^{2} c - 6 \, a c^{2}\right )} d x^{3} - {\left (2 \, b^{3} - 9 \, a b c\right )} d x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{2} x^{3} + 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d^{2} x^{2} - {\left (2 \, b^{3} - 9 \, a b c\right )} d^{2} x\right )} e^{4} + {\left (2 \, c^{3} d^{3} x^{3} - 9 \, b c^{2} d^{3} x^{2} - 9 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d^{3} x + {\left (2 \, b^{3} - 9 \, a b c\right )} d^{3}\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} - 3 \, b c^{2} d^{4} x - {\left (b^{2} c - 6 \, a c^{2}\right )} d^{4}\right )} e^{2} + 3 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\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 ) + 6 \, {\left (c^{3} d^{5} e + {\left (b^{2} c - 3 \, a c^{2}\right )} x^{3} e^{6} - {\left (b c^{2} d x^{3} - 3 \, {\left (b^{2} c - 3 \, a c^{2}\right )} d x^{2}\right )} e^{5} + {\left (c^{3} d^{2} x^{3} - 3 \, b c^{2} d^{2} x^{2} + 3 \, {\left (b^{2} c - 3 \, a c^{2}\right )} d^{2} x\right )} e^{4} + {\left (3 \, c^{3} d^{3} x^{2} - 3 \, b c^{2} d^{3} x + {\left (b^{2} c - 3 \, a c^{2}\right )} d^{3}\right )} e^{3} + {\left (3 \, c^{3} d^{4} x - b c^{2} d^{4}\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 (c^{3} d^{4} e^{2} - {\left (a b c x + 3 \, a^{2} c - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} x^{2}\right )} e^{6} - {\left (2 \, b c^{2} d x^{2} - 5 \, a b c d - 5 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d x\right )} e^{5} + {\left (2 \, c^{3} d^{2} x^{2} - 7 \, b c^{2} d^{2} x - 10 \, a c^{2} d^{2}\right )} e^{4} + {\left (6 \, c^{3} d^{3} x + b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} d^{7} e^{3} + a^{2} c x^{3} e^{10} - {\left (2 \, a b c d x^{3} - 3 \, a^{2} c d x^{2}\right )} e^{9} - {\left (6 \, a b c d^{2} x^{2} - 3 \, a^{2} c d^{2} x - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{3}\right )} e^{8} - {\left (2 \, b c^{2} d^{3} x^{3} + 6 \, a b c d^{3} x - a^{2} c d^{3} - 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{3} x^{2}\right )} e^{7} + {\left (c^{3} d^{4} x^{3} - 6 \, b c^{2} d^{4} x^{2} - 2 \, a b c d^{4} + 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x\right )} e^{6} + {\left (3 \, c^{3} d^{5} x^{2} - 6 \, b c^{2} d^{5} x + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{5}\right )} e^{5} + {\left (3 \, c^{3} d^{6} x - 2 \, b c^{2} d^{6}\right )} e^{4}\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 {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {7}{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 {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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