Integrand size = 28, antiderivative size = 373 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{884 c^4 \sqrt {a+b x+c x^2}} \]
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Time = 0.28 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {699, 706, 705, 704, 313, 227, 1213, 435} \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]
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Rule 227
Rule 313
Rule 435
Rule 699
Rule 704
Rule 705
Rule 706
Rule 1213
Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx}{34 c} \\ & = -\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2} \, dx}{884 c^2} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx}{5304 c^3} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{1768 c^3} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{1768 c^3 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{884 c^4 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (\left (b^2-4 a c\right )^{9/2} d^2 \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-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{9/2} d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{884 c^4 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{9/2} d^2 \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 {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{884 c^4 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.31 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{17} d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^3+\frac {\left (b^2-4 a c\right )^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1189\) vs. \(2(317)=634\).
Time = 4.49 (sec) , antiderivative size = 1190, normalized size of antiderivative = 3.19
method | result | size |
default | \(\text {Expression too large to display}\) | \(1190\) |
risch | \(\text {Expression too large to display}\) | \(2710\) |
elliptic | \(\text {Expression too large to display}\) | \(9950\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.93 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {3 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (1248 \, c^{8} d^{2} x^{7} + 4368 \, b c^{7} d^{2} x^{6} + 72 \, {\left (79 \, b^{2} c^{6} + 48 \, a c^{7}\right )} d^{2} x^{5} + 60 \, {\left (55 \, b^{3} c^{5} + 144 \, a b c^{6}\right )} d^{2} x^{4} + 4 \, {\left (187 \, b^{4} c^{4} + 1804 \, a b^{2} c^{5} + 712 \, a^{2} c^{6}\right )} d^{2} x^{3} + 6 \, {\left (b^{5} c^{3} + 364 \, a b^{3} c^{4} + 712 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 4 \, {\left (b^{6} c^{2} - 15 \, a b^{4} c^{3} - 486 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x + {\left (3 \, b^{7} c - 46 \, a b^{5} c^{2} + 260 \, a^{2} b^{3} c^{3} + 128 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{2652 \, c^{4}} \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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