Integrand size = 22, antiderivative size = 267 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \]
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Time = 0.14 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1675, 654, 626, 635, 212} \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=-\frac {5 \left (b^2-4 a c\right )^3 \left (-4 a c C+32 A c^2+9 b^2 C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{6144 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1675
Rubi steps \begin{align*} \text {integral}& = \frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (8 A c-a C-\frac {9 b C x}{2}\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c} \\ & = -\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (\frac {9 b^2 C}{2}+2 c (8 A c-a C)\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2} \\ & = \frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3} \\ & = -\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^4} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^5} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{16384 c^5} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \\ \end{align*}
Time = 3.66 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.35 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (224 A c^2 (b+2 c x) \left (15 b^4-40 b^3 c x+32 b c^2 x \left (13 a+8 c x^2\right )+8 b^2 c \left (-20 a+11 c x^2\right )+16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right )+C \left (945 b^7-630 b^6 c x+8 b^4 c^2 x \left (791 a-54 c x^2\right )+84 b^5 c \left (-125 a+6 c x^2\right )+16 b^3 c^2 \left (2359 a^2-284 a c x^2+24 c^2 x^4\right )+96 b^2 c^3 x \left (-199 a^2+36 a c x^2+648 c^2 x^4\right )+896 c^4 x \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )+64 b c^3 \left (-663 a^3+174 a^2 c x^2+2456 a c^2 x^4+1584 c^3 x^6\right )\right )\right )-105 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{344064 c^{11/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(237)=474\).
Time = 0.88 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.79
method | result | size |
risch | \(\frac {\left (43008 C \,c^{7} x^{7}+101376 C b \,c^{6} x^{6}+57344 A \,c^{7} x^{5}+121856 C a \,c^{6} x^{5}+62208 C \,b^{2} c^{5} x^{5}+143360 A b \,c^{6} x^{4}+157184 C a b \,c^{5} x^{4}+384 C \,b^{3} c^{4} x^{4}+186368 A a \,c^{6} x^{3}+96768 A \,b^{2} c^{5} x^{3}+105728 C \,a^{2} c^{5} x^{3}+3456 C a \,b^{2} c^{4} x^{3}-432 C \,b^{4} c^{3} x^{3}+279552 A a b \,c^{5} x^{2}+1792 A \,b^{3} c^{4} x^{2}+11136 C \,a^{2} b \,c^{4} x^{2}-4544 C a \,b^{3} c^{3} x^{2}+504 C \,b^{5} c^{2} x^{2}+236544 A \,a^{2} c^{5} x +21504 A a \,b^{2} c^{4} x -2240 A \,b^{4} c^{3} x +13440 C \,a^{3} c^{4} x -19104 C \,a^{2} b^{2} c^{3} x +6328 C a \,b^{4} c^{2} x -630 C \,b^{6} c x +118272 A \,a^{2} b \,c^{4}-35840 A a \,b^{3} c^{3}+3360 A \,b^{5} c^{2}-42432 C \,a^{3} b \,c^{3}+37744 C \,a^{2} b^{3} c^{2}-10500 C a \,b^{5} c +945 C \,b^{7}\right ) \sqrt {c \,x^{2}+b x +a}}{344064 c^{5}}+\frac {5 \left (2048 A \,a^{3} c^{5}-1536 A \,a^{2} b^{2} c^{4}+384 A a \,b^{4} c^{3}-32 A \,b^{6} c^{2}-256 C \,a^{4} c^{4}+768 C \,a^{3} b^{2} c^{3}-480 C \,a^{2} b^{4} c^{2}+112 C a \,b^{6} c -9 C \,b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32768 c^{\frac {11}{2}}}\) | \(479\) |
default | \(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+C \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )\) | \(487\) |
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Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (237) = 474\).
Time = 0.36 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.57 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\left [\frac {105 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c - 2048 \, A a^{3} c^{5} + 256 \, {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} c^{4} - 384 \, {\left (2 \, C a^{3} b^{2} + A a b^{4}\right )} c^{3} + 32 \, {\left (15 \, C a^{2} b^{4} + A b^{6}\right )} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (43008 \, C c^{8} x^{7} + 101376 \, C b c^{7} x^{6} + 945 \, C b^{7} c - 10500 \, C a b^{5} c^{2} + 118272 \, A a^{2} b c^{5} + 256 \, {\left (243 \, C b^{2} c^{6} + 476 \, C a c^{7} + 224 \, A c^{8}\right )} x^{5} - 64 \, {\left (663 \, C a^{3} b + 560 \, A a b^{3}\right )} c^{4} + 128 \, {\left (3 \, C b^{3} c^{5} + 1228 \, C a b c^{6} + 1120 \, A b c^{7}\right )} x^{4} + 112 \, {\left (337 \, C a^{2} b^{3} + 30 \, A b^{5}\right )} c^{3} - 16 \, {\left (27 \, C b^{4} c^{4} - 216 \, C a b^{2} c^{5} - 11648 \, A a c^{7} - 112 \, {\left (59 \, C a^{2} + 54 \, A b^{2}\right )} c^{6}\right )} x^{3} + 8 \, {\left (63 \, C b^{5} c^{3} - 568 \, C a b^{3} c^{4} + 34944 \, A a b c^{6} + 16 \, {\left (87 \, C a^{2} b + 14 \, A b^{3}\right )} c^{5}\right )} x^{2} - 2 \, {\left (315 \, C b^{6} c^{2} - 3164 \, C a b^{4} c^{3} - 118272 \, A a^{2} c^{6} - 1344 \, {\left (5 \, C a^{3} + 8 \, A a b^{2}\right )} c^{5} + 16 \, {\left (597 \, C a^{2} b^{2} + 70 \, A b^{4}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1376256 \, c^{6}}, \frac {105 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c - 2048 \, A a^{3} c^{5} + 256 \, {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} c^{4} - 384 \, {\left (2 \, C a^{3} b^{2} + A a b^{4}\right )} c^{3} + 32 \, {\left (15 \, C a^{2} b^{4} + A b^{6}\right )} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (43008 \, C c^{8} x^{7} + 101376 \, C b c^{7} x^{6} + 945 \, C b^{7} c - 10500 \, C a b^{5} c^{2} + 118272 \, A a^{2} b c^{5} + 256 \, {\left (243 \, C b^{2} c^{6} + 476 \, C a c^{7} + 224 \, A c^{8}\right )} x^{5} - 64 \, {\left (663 \, C a^{3} b + 560 \, A a b^{3}\right )} c^{4} + 128 \, {\left (3 \, C b^{3} c^{5} + 1228 \, C a b c^{6} + 1120 \, A b c^{7}\right )} x^{4} + 112 \, {\left (337 \, C a^{2} b^{3} + 30 \, A b^{5}\right )} c^{3} - 16 \, {\left (27 \, C b^{4} c^{4} - 216 \, C a b^{2} c^{5} - 11648 \, A a c^{7} - 112 \, {\left (59 \, C a^{2} + 54 \, A b^{2}\right )} c^{6}\right )} x^{3} + 8 \, {\left (63 \, C b^{5} c^{3} - 568 \, C a b^{3} c^{4} + 34944 \, A a b c^{6} + 16 \, {\left (87 \, C a^{2} b + 14 \, A b^{3}\right )} c^{5}\right )} x^{2} - 2 \, {\left (315 \, C b^{6} c^{2} - 3164 \, C a b^{4} c^{3} - 118272 \, A a^{2} c^{6} - 1344 \, {\left (5 \, C a^{3} + 8 \, A a b^{2}\right )} c^{5} + 16 \, {\left (597 \, C a^{2} b^{2} + 70 \, A b^{4}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{688128 \, c^{6}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 2691 vs. \(2 (270) = 540\).
Time = 0.58 (sec) , antiderivative size = 2691, normalized size of antiderivative = 10.08 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (237) = 474\).
Time = 0.30 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.80 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, C c^{2} x + 33 \, C b c\right )} x + \frac {243 \, C b^{2} c^{7} + 476 \, C a c^{8} + 224 \, A c^{9}}{c^{7}}\right )} x + \frac {3 \, C b^{3} c^{6} + 1228 \, C a b c^{7} + 1120 \, A b c^{8}}{c^{7}}\right )} x - \frac {27 \, C b^{4} c^{5} - 216 \, C a b^{2} c^{6} - 6608 \, C a^{2} c^{7} - 6048 \, A b^{2} c^{7} - 11648 \, A a c^{8}}{c^{7}}\right )} x + \frac {63 \, C b^{5} c^{4} - 568 \, C a b^{3} c^{5} + 1392 \, C a^{2} b c^{6} + 224 \, A b^{3} c^{6} + 34944 \, A a b c^{7}}{c^{7}}\right )} x - \frac {315 \, C b^{6} c^{3} - 3164 \, C a b^{4} c^{4} + 9552 \, C a^{2} b^{2} c^{5} + 1120 \, A b^{4} c^{5} - 6720 \, C a^{3} c^{6} - 10752 \, A a b^{2} c^{6} - 118272 \, A a^{2} c^{7}}{c^{7}}\right )} x + \frac {945 \, C b^{7} c^{2} - 10500 \, C a b^{5} c^{3} + 37744 \, C a^{2} b^{3} c^{4} + 3360 \, A b^{5} c^{4} - 42432 \, C a^{3} b c^{5} - 35840 \, A a b^{3} c^{5} + 118272 \, A a^{2} b c^{6}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c + 480 \, C a^{2} b^{4} c^{2} + 32 \, A b^{6} c^{2} - 768 \, C a^{3} b^{2} c^{3} - 384 \, A a b^{4} c^{3} + 256 \, C a^{4} c^{4} + 1536 \, A a^{2} b^{2} c^{4} - 2048 \, A a^{3} c^{5}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \]
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Timed out. \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\int \left (C\,x^2+A\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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