Integrand size = 30, antiderivative size = 469 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {826, 857, 732, 435, 430} \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {a+b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^2 \sqrt {d+e x}} \]
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Rule 430
Rule 435
Rule 732
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {\frac {1}{2} \left (8 b c d-3 b^2 e-4 a c e\right )+4 c (2 c d-b e) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 e^2} \\ & = \frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {(8 c (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 e^3}+\frac {\left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 e^3} \\ & = \frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 )}{3 e^3 \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 (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\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 )}{3 c e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 )}{3 e^3 \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 (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\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 )}{3 c e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.72 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.04 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 (8 c d-3 b e+2 c e x) (a+x (b+c x))}{e^2 (d+e x)}-\frac {(d+e x) \left (-\frac {16 e^2 (-2 c d+b 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}}} (a+x (b+c x))}{(d+e x)^2}-\frac {4 i \sqrt {2} (2 c d-b e) \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 e^2-4 a c e^2+8 c d \sqrt {\left (b^2-4 a c\right ) e^2}-4 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)}} \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 )}{e^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}}}}\right )}{3 \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(956\) vs. \(2(411)=822\).
Time = 3.34 (sec) , antiderivative size = 957, normalized size of antiderivative = 2.04
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +a e \right ) \left (b e -2 c d \right )}{e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {4 c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 e^{2}}+\frac {2 \left (\frac {2 a c \,e^{2}+b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}}{e^{3}}-\frac {\left (b e -2 c d \right ) \left (b e -c d \right )}{e^{3}}+\frac {b \left (b e -2 c d \right )}{e^{2}}-\frac {4 c \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 e^{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 (\frac {c \left (3 b e -2 c d \right )}{e^{2}}+\frac {c \left (b e -2 c d \right )}{e^{2}}-\frac {4 c \left (b e +c d \right )}{3 e^{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}}}\, \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 )}{\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}}\) | \(957\) |
risch | \(\text {Expression too large to display}\) | \(2542\) |
default | \(\text {Expression too large to display}\) | \(2977\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.07 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (16 \, c^{2} d^{3} - 16 \, b c d^{2} e + {\left (b^{2} + 12 \, a c\right )} d e^{2} + {\left (16 \, c^{2} d^{2} e - 16 \, b c d e^{2} + {\left (b^{2} + 12 \, a c\right )} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left (2 \, c^{2} d^{2} e - b c d e^{2} + {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (2 \, c^{2} e^{3} x + 8 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{9 \, {\left (c e^{5} x + c d e^{4}\right )}} \]
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\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (b + 2 c x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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