Integrand size = 29, antiderivative size = 460 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (b B-2 A c) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} (b B d-2 A c d+A b e-2 a B e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {834, 857, 732, 435, 430} \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b B-2 A c) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {d+e x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 430
Rule 435
Rule 732
Rule 834
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-2 a B-(b B-2 A c) x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {1}{2} (A b-2 a B) e+\frac {1}{2} (b B-2 A c) e x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2 (A b-2 a B-(b B-2 A c) x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {(b B-2 A c) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}+\frac {(b B d-2 A c d+A b e-2 a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2 (A b-2 a B-(b B-2 A c) x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {2} (b B-2 A c) \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 \sqrt {b^2-4 a c} \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} (b B d-2 A c d+A b e-2 a 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 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 (A b-2 a B-(b B-2 A c) x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (b B-2 A c) \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 \sqrt {b^2-4 a c} \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} (b B d-2 A c d+A b e-2 a 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 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 27.96 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {4 (B (2 a+b x)-A (b+2 c x))}{b^2-4 a c}+\frac {(d+e x) \left (\frac {4 (b B-2 A c) e^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}}} (a+x (b+c x))}{(d+e x)^2}-\frac {i \sqrt {2} (b B-2 A c) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \text {arcsinh}\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^2 B e-4 a B c e-b B \sqrt {\left (b^2-4 a c\right ) e^2}+2 A c \sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\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 )}{c \left (-b^2+4 a c\right ) e \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 )}{2 \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(950\) vs. \(2(408)=816\).
Time = 1.28 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.07
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (x c e +c d \right ) \left (-\frac {\left (2 A c -B b \right ) x}{c \left (4 a c -b^{2}\right )}-\frac {A b -2 B a}{c \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (x c e +c d \right )}}+\frac {2 \left (\frac {B e}{c}+\frac {4 A \,c^{2} d -4 B a c e +b^{2} B e -2 B b c d}{\left (4 a c -b^{2}\right ) c}-\frac {e \left (A b -2 B a \right )}{4 a c -b^{2}}-\frac {2 d \left (2 A c -B b \right )}{4 a c -b^{2}}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\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}}}\, \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}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \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 +b d x +a d}}-\frac {2 \left (2 A c -B b \right ) e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\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}}}\, \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 ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \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 ) F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \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 )}{\left (4 a c -b^{2}\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}}\) | \(951\) |
default | \(\text {Expression too large to display}\) | \(3171\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d + {\left (B b^{2} c - {\left (6 \, B a - A b\right )} c^{2}\right )} e\right )} x^{2} + {\left (B a b c - 2 \, A a c^{2}\right )} d + {\left (B a b^{2} - {\left (6 \, B a^{2} - A a b\right )} c\right )} e + {\left ({\left (B b^{2} c - 2 \, A b c^{2}\right )} d + {\left (B b^{3} - {\left (6 \, B a b - A b^{2}\right )} c\right )} e\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} e x^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} e x + {\left (B a b c - 2 \, A a c^{2}\right )} e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{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 )}}{3 \, c^{2} e^{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 )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left ({\left (2 \, B a - A b\right )} c^{2} e + {\left (B b c^{2} - 2 \, A c^{3}\right )} e x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} e\right )}} \]
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\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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