Integrand size = 24, antiderivative size = 629 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.51 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {758, 848, 857, 732, 435, 430} \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{15 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]
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Rule 430
Rule 435
Rule 732
Rule 758
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} (-5 c d+4 b e)+\frac {3 c e x}{2}}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {1}{4} \left (15 c^2 d^2+8 b^2 e^2-c e (19 b d+9 a e)\right )-c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} c \left (15 c^2 d^3+4 b e^2 (b d+a e)-c d e (11 b d+17 a e)\right )-\frac {1}{8} c e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {(4 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (c \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 )}{15 \left (c d^2-b d e+a e^2\right )^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 (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-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 )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 )}{15 \left (c d^2-b d e+a e^2\right )^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 {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-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 )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 30.09 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right ) \left (-\frac {2 e}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {8 e (-2 c d+b e)}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {2 e \left (-23 c^2 d^2+23 b c d e-8 b^2 e^2+9 a c e^2\right )}{15 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}\right )}{\sqrt {a+x (b+c x)}}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (\left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 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 {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) 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 )+\left (-30 c^3 d^3+8 b^2 e^2 \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )-c^2 d \left (-45 b d e-34 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 b^2 d e-17 a b e^2+23 b d \sqrt {\left (b^2-4 a c\right ) e^2}+9 a e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \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 )\right )}{2 \sqrt {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}}} \sqrt {d+e x}}\right )}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+x (b+c x)} \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. \(1157\) vs. \(2(559)=1118\).
Time = 2.06 (sec) , antiderivative size = 1158, normalized size of antiderivative = 1.84
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1158\) |
default | \(\text {Expression too large to display}\) | \(14312\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 1403, normalized size of antiderivative = 2.23 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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