Optimal. Leaf size=271 \[ \frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {698, 699, 705,
704, 313, 227, 1213, 435} \begin {gather*} \frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}+\frac {3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 313
Rule 435
Rule 698
Rule 699
Rule 704
Rule 705
Rule 1213
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{3/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}+\frac {3 \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx}{2 c d^2}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2 d^2}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}-\frac {\left (3 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{20 c^2 d^2 \sqrt {a+b x+c x^2}}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}-\frac {\left (3 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{10 c^3 d^3 \sqrt {a+b x+c x^2}}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}+\frac {\left (3 \left (b^2-4 a c\right )^{3/2} \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-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{10 c^3 d^2 \sqrt {a+b x+c x^2}}-\frac {\left (3 \left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{10 c^3 d^2 \sqrt {a+b x+c x^2}}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}-\frac {\left (3 \left (b^2-4 a c\right )^{3/2} \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 {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{10 c^3 d^2 \sqrt {a+b x+c x^2}}\\ &=\frac {3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{10 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt {b d+2 c d x}}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 99, normalized size = 0.37 \begin {gather*} \frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt {d (b+2 c x)} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs.
\(2(229)=458\).
time = 0.85, size = 504, normalized size = 1.86
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (48 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a^{2} c^{2}-24 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2} c +3 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{4}+4 c^{4} x^{4}+8 b \,c^{3} x^{3}-16 x^{2} c^{3} a +10 b^{2} c^{2} x^{2}-16 x a b \,c^{2}+6 b^{3} c x -20 a^{2} c^{2}+6 a c \,b^{2}\right )}{20 d^{2} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right ) c^{3}}\) | \(504\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (2 c^{2} d \,x^{2}+2 x b c d +2 a c d \right ) \left (4 a c -b^{2}\right )}{8 c^{3} d^{2} \sqrt {\left (x +\frac {b}{2 c}\right ) \left (2 c^{2} d \,x^{2}+2 x b c d +2 a c d \right )}}+\frac {x \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{10 d^{2} c}+\frac {b \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{20 d^{2} c^{2}}+\frac {2 \left (\frac {b \left (8 a c -b^{2}\right )}{16 c^{2} d}+\frac {b \left (4 a c -b^{2}\right )}{16 c^{2} d}-\frac {a b}{10 c d}-\frac {b \left (a c d +\frac {1}{2} b^{2} d \right )}{20 d^{2} c^{2}}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{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 {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}+\frac {2 \left (\frac {8 a c +b^{2}}{8 c d}+\frac {4 a c -b^{2}}{8 c d}-\frac {3 a c d +\frac {3}{2} b^{2} d}{10 d^{2} c}-\frac {3 b^{2}}{20 d c}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{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 {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )-\frac {b \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{2 c}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(1229\) |
risch | \(\text {Expression too large to display}\) | \(2891\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.93, size = 150, normalized size = 0.55 \begin {gather*} \frac {3 \, \sqrt {2} {\left (b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (2 \, c^{3} x^{2} + 2 \, b c^{2} x + 3 \, b^{2} c - 10 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{10 \, {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 \left (b + 2 c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 (b\,d+2\,c\,d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________