Integrand size = 28, antiderivative size = 247 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {520 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{7 \sqrt {a+b x+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {700, 706, 705, 703, 227} \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{7 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}+\frac {624}{7} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 227
Rule 700
Rule 703
Rule 705
Rule 706
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (26 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\left (156 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{7} \left (780 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{7} \left (260 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (260 c^2 \left (b^2-4 a c\right )^2 d^8 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\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}{7 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (520 c \left (b^2-4 a c\right )^2 d^7 \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 )}{7 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1040}{7} c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {624}{7} c^2 d^5 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {520 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \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 )}{7 \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 = 10.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.02 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 d^7 \sqrt {d (b+2 c x)} \left (-7 b^6-266 b^5 c x+16 b^3 c^2 x \left (221 a+112 c x^2\right )+2 b^4 c \left (-91 a+219 c x^2\right )+32 b c^3 x \left (-273 a^2-104 a c x^2+36 c^2 x^4\right )+16 b^2 c^2 \left (156 a^2+117 a c x^2+116 c^2 x^4\right )-32 c^3 \left (195 a^3+273 a^2 c x^2+52 a c^2 x^4-12 c^3 x^6\right )+780 c \left (b^2-4 a c\right )^2 (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{21 (a+x (b+c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1472\) vs. \(2(207)=414\).
Time = 10.74 (sec) , antiderivative size = 1473, normalized size of antiderivative = 5.96
method | result | size |
default | \(\text {Expression too large to display}\) | \(1473\) |
risch | \(\text {Expression too large to display}\) | \(1526\) |
elliptic | \(\text {Expression too large to display}\) | \(1532\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.72 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (390 \, \sqrt {2} {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{7} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{7} x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d^{7}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (384 \, c^{6} d^{7} x^{6} + 1152 \, b c^{5} d^{7} x^{5} + 64 \, {\left (29 \, b^{2} c^{4} - 26 \, a c^{5}\right )} d^{7} x^{4} + 256 \, {\left (7 \, b^{3} c^{3} - 13 \, a b c^{4}\right )} d^{7} x^{3} + 6 \, {\left (73 \, b^{4} c^{2} + 312 \, a b^{2} c^{3} - 1456 \, a^{2} c^{4}\right )} d^{7} x^{2} - 2 \, {\left (133 \, b^{5} c - 1768 \, a b^{3} c^{2} + 4368 \, a^{2} b c^{3}\right )} d^{7} x - {\left (7 \, b^{6} + 182 \, a b^{4} c - 2496 \, a^{2} b^{2} c^{2} + 6240 \, a^{3} c^{3}\right )} d^{7}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{21 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{15/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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