Integrand size = 23, antiderivative size = 474 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.40 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {746, 824, 826, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 d e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 c \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{21 d e^5 \sqrt {d+e x} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{21 d e^3 (d+e x)^{5/2} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 746
Rule 824
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e} \\ & = -\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} b c d (16 c d-13 b e)-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx}{7 d e^3 (c d-b e)} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 d e^5 (c d-b e)} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{21 d e^6 (c d-b e)}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 e^6} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{21 d e^6 (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{21 e^6 \sqrt {b x+c x^2}} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.58 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.05 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (3 b^3 e^6 x^3-b^2 c d e^2 \left (51 d^3+169 d^2 e x+194 d e^2 x^2+85 e^3 x^3\right )-c^3 d^2 \left (128 d^4+416 d^3 e x+464 d^2 e^2 x^2+186 d e^3 x^3+7 e^4 x^4\right )+b c^2 d e \left (176 d^4+576 d^3 e x+649 d^2 e^2 x^2+265 d e^3 x^3+7 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} c (d+e x)^3 \left (\sqrt {\frac {b}{c}} \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (128 c^3 d^3-208 b c^2 d^2 e+83 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{21 b d e^6 (c d-b e) x^3 (b+c x)^3 (d+e x)^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(414)=828\).
Time = 3.04 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.89
method | result | size |
elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 e^{9} \left (x +\frac {d}{e}\right )^{4}}+\frac {6 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 e^{8} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{21 e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 d \left (b e -c d \right ) e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{5}}+\frac {2 \left (\frac {c \left (3 b^{2} e^{2}-12 b c d e +10 c^{2} d^{2}\right )}{e^{6}}-\frac {c \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right )}{21 e^{6}}+\frac {3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}}{21 e^{6} d}-\frac {b \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {c^{2} b d}{3 e^{5}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c^{2} \left (3 b e -4 c d \right )}{e^{5}}-\frac {c \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {2 c^{2} \left (b e +c d \right )}{3 e^{5}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(894\) |
default | \(\text {Expression too large to display}\) | \(3284\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 1184, normalized size of antiderivative = 2.50 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
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