Optimal. Leaf size=749 \[ -\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )-c e \left (2 a e (9 C d-5 B e)-b \left (32 C d^2-5 e (5 B d-3 A e)\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.90, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1664, 828, 857,
732, 435, 430} \begin {gather*} \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}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e \left (-2 a e (9 C d-5 B e)-5 b e (5 B d-3 A e)+32 b C d^2\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\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^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\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 (c e (-6 a C e-5 b B e+8 b C d)-\left (c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^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^2 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 {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^3 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 828
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \int \frac {\left (-\frac {3 b C d^2-b e (3 B d-2 A e)+e (A c d-a C d+a B e)}{2 e}+\frac {1}{2} \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {\left (c d^2-b d e+a e^2\right ) \left (b^2 C d e+a b C e^2+2 a c e (6 C d-5 B e)-b c \left (24 C d^2-5 e (4 B d-3 A e)\right )\right )}{4 e}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) x}{4 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c e^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c e^4}+\frac {\left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c e^4}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\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^2 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 (2 \sqrt {2} \sqrt {b^2-4 a c} \left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\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^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\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^2 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 {2 \sqrt {2} \sqrt {b^2-4 a c} \left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right )\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^2 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 = 32.90, size = 13240, normalized size = 17.68 \begin {gather*} \text {Result too large to show} \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. \(8220\) vs.
\(2(687)=1374\).
time = 0.18, size = 8221, normalized size = 10.98
method | result | size |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +a e \right ) \left (A \,e^{2}-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 +b d x +a d}}{5 e^{2}}+\frac {2 \left (\frac {c e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{5 e^{2}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (\frac {A b \,e^{3}-d \,e^{2} c A +B a \,e^{3}-B b d \,e^{2}+B c \,d^{2} e -C a d \,e^{2}+C b \,d^{2} e -C c \,d^{3}}{e^{4}}-\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (e b -c d \right )}{e^{4}}+\frac {b \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e^{3}}-\frac {2 a d C}{5 e^{2}}-\frac {2 \left (\frac {c e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{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 {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\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 +b d x +a d}}+\frac {2 \left (\frac {A c \,e^{2}+B b \,e^{2}-B c d e +a C \,e^{2}-C b d e +C c \,d^{2}}{e^{3}}+\frac {\left (A \,e^{2}-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 e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{5 e^{2}}\right ) \left (e b +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 {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}}\, \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 {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\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 {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\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 +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\) | \(1215\) |
risch | \(\text {Expression too large to display}\) | \(1864\) |
default | \(\text {Expression too large to display}\) | \(8221\) |
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 = 0.12, size = 790, normalized size = 1.05 \begin {gather*} -\frac {2 \, {\left ({\left (48 \, C c^{3} d^{4} - {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} x e^{4} - {\left ({\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d x + {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} d\right )} e^{3} - {\left (8 \, {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} d^{2} x + {\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d^{2}\right )} e^{2} + 8 \, {\left (6 \, C c^{3} d^{3} x - {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} 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 (48 \, C c^{3} d^{3} e - {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} x e^{4} - {\left (8 \, {\left (C b c^{2} + 5 \, B c^{3}\right )} d x + {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} d\right )} e^{3} + 8 \, {\left (6 \, C c^{3} d^{2} x - {\left (C b c^{2} + 5 \, B c^{3}\right )} 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 (24 \, C c^{3} d^{2} e^{2} - {\left (3 \, C c^{3} x^{2} - 15 \, A c^{3} + {\left (C b c^{2} + 5 \, B c^{3}\right )} x\right )} e^{4} + {\left (6 \, C c^{3} d x - {\left (C b c^{2} + 20 \, B c^{3}\right )} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} x e^{6} + c^{3} 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 ) \sqrt {a + b x + c x^{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 )\,\sqrt {c\,x^2+b\,x+a}}{{\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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