Integrand size = 20, antiderivative size = 133 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 (b B-2 A c) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {128 c (b B-2 A c) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {652, 628, 627} \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {128 c (b+2 c x) (b B-2 A c)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}-\frac {16 (b+2 c x) (b B-2 A c)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]
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Rule 627
Rule 628
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-2 a B-(b B-2 A c) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {(8 (b B-2 A c)) \int \frac {1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 \left (b^2-4 a c\right )} \\ & = -\frac {2 (A b-2 a B-(b B-2 A c) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 (b B-2 A c) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {(64 c (b B-2 A c)) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2} \\ & = -\frac {2 (A b-2 a B-(b B-2 A c) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 (b B-2 A c) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {128 c (b B-2 A c) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 2.51 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {-2 A (b+2 c x) \left (3 b^4-16 b^3 c x+64 b c^2 x \left (5 a+4 c x^2\right )+8 b^2 c \left (-5 a+14 c x^2\right )+16 c^2 \left (15 a^2+20 a c x^2+8 c^2 x^4\right )\right )+2 B \left (96 a^3 c^2+48 a^2 b c (b+5 c x)+2 a b \left (-b^3+60 b^2 c x+240 b c^2 x^2+160 c^3 x^3\right )+b x \left (-5 b^4+40 b^3 c x+240 b^2 c^2 x^2+320 b c^3 x^3+128 c^4 x^4\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(121)=242\).
Time = 0.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94
method | result | size |
default | \(A \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )+B \left (-\frac {1}{5 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {b \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )\) | \(258\) |
trager | \(\frac {\frac {512}{15} A \,c^{5} x^{5}-\frac {256}{15} B b \,c^{4} x^{5}+\frac {256}{3} A b \,c^{4} x^{4}-\frac {128}{3} B \,b^{2} c^{3} x^{4}+\frac {256}{3} A a \,c^{4} x^{3}+64 A \,b^{2} c^{3} x^{3}-\frac {128}{3} B a b \,c^{3} x^{3}-32 B \,b^{3} c^{2} x^{3}+128 A a b \,c^{3} x^{2}+\frac {32}{3} A \,b^{3} c^{2} x^{2}-64 B a \,b^{2} c^{2} x^{2}-\frac {16}{3} B \,b^{4} c \,x^{2}+64 A \,a^{2} c^{3} x +32 A a \,b^{2} c^{2} x -\frac {4}{3} A \,b^{4} c x -32 B \,a^{2} b \,c^{2} x -16 B a \,b^{3} c x +\frac {2}{3} B \,b^{5} x +32 A \,a^{2} b \,c^{2}-\frac {16}{3} A a \,b^{3} c +\frac {2}{5} A \,b^{5}-\frac {64}{5} B \,a^{3} c^{2}-\frac {32}{5} B \,a^{2} b^{2} c +\frac {4}{15} B a \,b^{4}}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}\) | \(266\) |
gosper | \(\frac {\frac {512}{15} A \,c^{5} x^{5}-\frac {256}{15} B b \,c^{4} x^{5}+\frac {256}{3} A b \,c^{4} x^{4}-\frac {128}{3} B \,b^{2} c^{3} x^{4}+\frac {256}{3} A a \,c^{4} x^{3}+64 A \,b^{2} c^{3} x^{3}-\frac {128}{3} B a b \,c^{3} x^{3}-32 B \,b^{3} c^{2} x^{3}+128 A a b \,c^{3} x^{2}+\frac {32}{3} A \,b^{3} c^{2} x^{2}-64 B a \,b^{2} c^{2} x^{2}-\frac {16}{3} B \,b^{4} c \,x^{2}+64 A \,a^{2} c^{3} x +32 A a \,b^{2} c^{2} x -\frac {4}{3} A \,b^{4} c x -32 B \,a^{2} b \,c^{2} x -16 B a \,b^{3} c x +\frac {2}{3} B \,b^{5} x +32 A \,a^{2} b \,c^{2}-\frac {16}{3} A a \,b^{3} c +\frac {2}{5} A \,b^{5}-\frac {64}{5} B \,a^{3} c^{2}-\frac {32}{5} B \,a^{2} b^{2} c +\frac {4}{15} B a \,b^{4}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) | \(288\) |
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Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (121) = 242\).
Time = 2.87 (sec) , antiderivative size = 543, normalized size of antiderivative = 4.08 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (2 \, B a b^{4} + 3 \, A b^{5} - 128 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} x^{5} - 320 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} x^{4} - 80 \, {\left (3 \, B b^{3} c^{2} - 8 \, A a c^{4} + 2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} c^{3}\right )} x^{3} - 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \, {\left (B b^{4} c - 24 \, A a b c^{3} + 2 \, {\left (6 \, B a b^{2} - A b^{3}\right )} c^{2}\right )} x^{2} - 8 \, {\left (6 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} c + 5 \, {\left (B b^{5} + 96 \, A a^{2} c^{3} - 48 \, {\left (B a^{2} b - A a b^{2}\right )} c^{2} - 2 \, {\left (12 \, B a b^{3} + A b^{4}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (a^{3} b^{6} - 12 \, a^{4} b^{4} c + 48 \, a^{5} b^{2} c^{2} - 64 \, a^{6} c^{3} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{6} + 3 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{4} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (121) = 242\).
Time = 0.29 (sec) , antiderivative size = 433, normalized size of antiderivative = 3.26 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (3 \, B b^{3} c^{2} + 4 \, B a b c^{3} - 6 \, A b^{2} c^{3} - 8 \, A a c^{4}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (B b^{4} c + 12 \, B a b^{2} c^{2} - 2 \, A b^{3} c^{2} - 24 \, A a b c^{3}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} - 24 \, B a b^{3} c - 2 \, A b^{4} c - 48 \, B a^{2} b c^{2} + 48 \, A a b^{2} c^{2} + 96 \, A a^{2} c^{3}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} + 3 \, A b^{5} - 48 \, B a^{2} b^{2} c - 40 \, A a b^{3} c - 96 \, B a^{3} c^{2} + 240 \, A a^{2} b c^{2}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \]
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Time = 12.01 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.96 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {\frac {b\,c\,\left (256\,A\,c^2-128\,B\,b\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,c^2\,x\,\left (256\,A\,c^2-128\,B\,b\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}}+\frac {x\,\left (\frac {4\,A\,c^2}{5\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {2\,B\,b\,c}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )+\frac {2\,A\,b\,c}{5\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {4\,B\,a\,c}{5\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {x\,\left (\frac {2\,c^2\,\left (32\,A\,c-20\,B\,b\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {8\,B\,b\,c^2}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,c\,\left (32\,A\,c-20\,B\,b\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,B\,a\,c^2}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}-\frac {4\,B}{\left (60\,a\,c-15\,b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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