Integrand size = 24, antiderivative size = 581 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.57 (sec) , antiderivative size = 581, 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 = {756, 828, 857, 732, 435, 430} \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
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Rule 430
Rule 435
Rule 732
Rule 756
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (7 c d^2-e (3 b d+a e)\right )+2 e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 c} \\ & = \frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {4 \int \frac {-\frac {1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\right )\right )+\frac {1}{4} e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 e^2} \\ & = \frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 e^2}+-\frac {\left (4 \left (-\frac {1}{4} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac {1}{4} e^2 \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 e^3} \\ & = \frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{105 c^3 e^2 \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 (8 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {1}{4} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac {1}{4} e^2 \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\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 )}{105 c^3 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{105 c^3 e^2 \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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 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 )}{105 c^3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 34.07 (sec) , antiderivative size = 1289, normalized size of antiderivative = 2.22 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\sqrt {d+e x} \left (\frac {2 \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+10 a c e^2\right )}{105 c^2 e}+\frac {2 (8 c d+b e) x}{35 c}+\frac {2 e x^2}{7}\right ) \sqrt {a+x (b+c x)}+\frac {(d+e x)^{3/2} \sqrt {a+x (b+c x)} \left (4 (-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}}} \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \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 (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 (-8 b^4 e^4+b^3 e^3 \left (27 c d+8 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-b^2 c e^2 \left (27 c d^2-37 a e^2+19 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+2 c^2 \left (-10 a^2 e^4-3 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (54 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+b c e \left (9 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-a e^2 \left (108 c d+29 \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 )}{210 c^3 e^3 \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}}} \sqrt {a+b x+c x^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}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1211\) vs. \(2(517)=1034\).
Time = 1.26 (sec) , antiderivative size = 1212, normalized size of antiderivative = 2.09
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1212\) |
risch | \(\text {Expression too large to display}\) | \(2640\) |
default | \(\text {Expression too large to display}\) | \(6517\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.02 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (6 \, c^{4} d^{4} - 12 \, b c^{3} d^{3} e - {\left (17 \, b^{2} c^{2} - 104 \, a c^{3}\right )} d^{2} e^{2} + {\left (23 \, b^{3} c - 104 \, a b c^{2}\right )} d e^{3} - {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (6 \, c^{4} d^{3} e - 9 \, b c^{3} d^{2} e^{2} + {\left (19 \, b^{2} c^{2} - 58 \, a c^{3}\right )} d e^{3} - {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 3 \, c^{4} d^{2} e^{2} + 9 \, b c^{3} d e^{3} - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} e^{4} + 3 \, {\left (8 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{4} e^{3}} \]
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\[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int \left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}\, dx \]
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\[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int {\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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