Optimal. Leaf size=508 \[ -\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (C e (b d-a e)-c \left (2 C d^2-e (B d-A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 C d-B e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.41, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1664, 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 (-C e (b d-a e)-c e (B d-A e)+2 c C 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 )}{c e^2 \sqrt {a+b x+c x^2} \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 )}}}-\frac {2 \sqrt {a+b x+c x^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 )}-\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 )}{c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 732
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \int \frac {-\frac {b d (C d-B e)+e (A c d-a C d+a B e)}{2 e}+\frac {1}{2} \left (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C e\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{e^2}-\frac {\left (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C e\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^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 (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C 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 )}{c 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 {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 )}{c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^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 (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C 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 )}{c 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 {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 )}{c e^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 = 24.69, size = 772, normalized size = 1.52 \begin {gather*} \frac {2 \left (-e^2 \left (C d^2+e (-B d+A e)\right ) (a+x (b+c x))+\frac {e^2 \left (2 c C d^2+C e (-b d+a e)+c e (-B d+A e)\right ) (a+x (b+c x))}{c}-\frac {i (d+e x)^{3/2} \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 (2 c C d^2+C e (-b d+a e)+c e (-B d+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 (-b^2 C d e^2+2 a c C d e^2-2 a B c e^3-2 c C d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+B c d e \sqrt {\left (b^2-4 a c\right ) e^2}-a C e^2 \sqrt {\left (b^2-4 a c\right ) e^2}-A c e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b \left (B c d e^2+A c e^3+a C e^3+C d 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} 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}}}}\right )}{e^3 \left (c d^2+e (-b d+a e)\right ) \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. \(6052\) vs.
\(2(456)=912\).
time = 0.17, size = 6053, normalized size = 11.92
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^{2} \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 \left (\frac {B e -C d}{e^{2}}-\frac {\left (e b -c d \right ) \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e^{2} \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {b \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e \left (a \,e^{2}-d e b +c \,d^{2}\right )}\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 {C}{e}+\frac {c \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e \left (a \,e^{2}-d e b +c \,d^{2}\right )}\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}}\) | \(965\) |
default | \(\text {Expression too large to display}\) | \(6053\) |
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 = 0.10, size = 687, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, C c^{2} d^{4} + {\left (C a b - {\left (3 \, B a - A b\right )} c\right )} x e^{4} - {\left ({\left (C b^{2} + 2 \, A c^{2} - 2 \, {\left (2 \, C a + B b\right )} c\right )} d x - {\left (C a b - {\left (3 \, B a - A b\right )} c\right )} d\right )} e^{3} - {\left ({\left (2 \, C b c + B c^{2}\right )} d^{2} x + {\left (C b^{2} + 2 \, A c^{2} - 2 \, {\left (2 \, C a + B b\right )} c\right )} d^{2}\right )} e^{2} + {\left (2 \, C c^{2} d^{3} x - {\left (2 \, C b c + B c^{2}\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 (2 \, C c^{2} d^{3} e + {\left (C a c + A c^{2}\right )} x e^{4} - {\left ({\left (C b c + B c^{2}\right )} d x - {\left (C a c + A c^{2}\right )} d\right )} e^{3} + {\left (2 \, C c^{2} d^{2} x - {\left (C b c + B c^{2}\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 (C c^{2} d^{2} e^{2} - B c^{2} d e^{3} + A c^{2} e^{4}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{3 \, {\left (c^{3} d^{3} e^{3} + a c^{2} x e^{6} - {\left (b c^{2} d x - a c^{2} d\right )} e^{5} + {\left (c^{3} d^{2} x - b c^{2} d^{2}\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 {A + B x + C x^{2}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{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 {C\,x^2+B\,x+A}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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