Integrand size = 24, antiderivative size = 578 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (8 c^2 d^3+a b e^3-c d e (7 b d-4 a e)+e \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.36 (sec) , antiderivative size = 578, 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 = {746, 824, 857, 732, 435, 430} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{5 e^3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]
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Rule 430
Rule 435
Rule 732
Rule 746
Rule 824
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx}{5 e} \\ & = -\frac {2 \left (8 c^2 d^3+a b e^3-c d e (7 b d-4 a e)+e \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} c \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )-\frac {1}{2} c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{5 e^3 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \left (8 c^2 d^3+a b e^3-c d e (7 b d-4 a e)+e \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{5 e^4}+\frac {\left (c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{5 e^4 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \left (8 c^2 d^3+a b e^3-c d e (7 b d-4 a e)+e \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 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 )}{5 e^4 \left (c d^2-b d e+a e^2\right ) \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 (16 \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 )}{5 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \left (8 c^2 d^3+a b e^3-c d e (7 b d-4 a e)+e \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 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 )}{5 e^4 \left (c d^2-b d e+a e^2\right ) \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 {16 \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 )}{5 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 32.34 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e x} (a+x (b+c x))^{3/2} \left (-\frac {2 \left (c d^2-b d e+a e^2\right )}{5 e^3 (d+e x)^3}+\frac {4 (2 c d-b e)}{5 e^3 (d+e x)^2}-\frac {2 \left (11 c^2 d^2-11 b c d e+b^2 e^2+7 a c e^2\right )}{5 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)}\right )}{a+b x+c x^2}-\frac {(d+e x)^{3/2} (a+x (b+c x))^{3/2} \left (-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}}} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 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} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 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^3 e^3+b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-2 c d+3 \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 )}{10 e^5 \left (c d^2-b d e+a e^2\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 (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}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1152\) vs. \(2(514)=1028\).
Time = 1.08 (sec) , antiderivative size = 1153, normalized size of antiderivative = 1.99
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1153\) |
default | \(\text {Expression too large to display}\) | \(12944\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1032, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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