Integrand size = 24, antiderivative size = 579 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c^2 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
[Out]
Time = 0.52 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {748, 828, 857, 732, 435, 430} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c^2 e^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (11 b d-10 a e)+b^2 e^2-3 c e x (2 c d-b e)+8 c^2 d^2\right )}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e} \]
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 748
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}-\frac {3 \int \frac {(b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e} \\ & = \frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}+\frac {2 \int \frac {\frac {1}{2} \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\right )-(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 c e^3} \\ & = \frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}-\frac {\left (2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{35 c e^4}+\frac {\left (2 \left (d (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {1}{2} e \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 c e^4} \\ & = \frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a 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 )}{35 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 (4 \sqrt {2} \sqrt {b^2-4 a c} \left (d (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {1}{2} e \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\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 )}{35 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a 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 )}{35 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 (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-16 b c d e-b^2 e^2+20 a c e^2\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 )}{35 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 33.78 (sec) , antiderivative size = 1296, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (8 c^2 d^2-11 b c d e+b^2 e^2+15 a c e^2\right )}{35 c e^3}+\frac {4 (-3 c d+4 b e) x}{35 e^2}+\frac {2 c x^2}{7 e}\right ) (a+x (b+c x))^{3/2}}{a+b x+c x^2}+\frac {(d+e x)^{3/2} (a+x (b+c x))^{3/2} \left (-4 (-2 c d+b e) \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) \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}}} \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )+\frac {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 ) \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} 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^4 e^4+b^3 e^3 \left (-c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b^2 c e^2 \left (c d^2+9 a e^2+2 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-4 b c e \left (3 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-c d+2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+4 c^2 \left (-5 a^2 e^4+2 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (-c d+4 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \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 )}{35 c^2 e^5 \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}}} \left (a+b x+c x^2\right )^{3/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1181\) vs. \(2(515)=1030\).
Time = 1.00 (sec) , antiderivative size = 1182, normalized size of antiderivative = 2.04
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1182\) |
risch | \(\text {Expression too large to display}\) | \(2639\) |
default | \(\text {Expression too large to display}\) | \(6516\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left ({\left (16 \, c^{4} d^{4} - 32 \, b c^{3} d^{3} e + {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{2} e^{2} + {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d e^{3} + {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c - 8 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (5 \, c^{4} e^{4} x^{2} + 8 \, c^{4} d^{2} e^{2} - 11 \, b c^{3} d e^{3} + {\left (b^{2} c^{2} + 15 \, a c^{3}\right )} e^{4} - 2 \, {\left (3 \, c^{4} d e^{3} - 4 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{105 \, c^{3} e^{5}} \]
[In]
[Out]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\sqrt {d + e x}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{\sqrt {d+e\,x}} \,d x \]
[In]
[Out]