Integrand size = 23, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {1}{2} \sqrt {c} (3 b B+2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
[Out]
Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {824, 826, 857, 635, 212, 738} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=-\frac {\left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {1}{2} \sqrt {c} (2 A c+3 b B) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )+\frac {\sqrt {a+b x+c x^2} \left (2 c x (6 a B+A b)-8 a A c-6 a b B+A b^2\right )}{8 a x}-\frac {\left (a+b x+c x^2\right )^{3/2} (3 x (2 a B+A b)+4 a A)}{12 a x^3} \]
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 824
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\int \frac {\left (\frac {1}{2} \left (-6 a b B+A \left (b^2-8 a c\right )\right )-(A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{x^2} \, dx}{4 a} \\ & = \frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+\frac {\int \frac {\frac {1}{2} \left (6 a B \left (b^2+4 a c\right )-2 A \left (\frac {b^3}{2}-6 a b c\right )\right )+4 a c (3 b B+2 A c) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 a} \\ & = \frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+\frac {1}{2} (c (3 b B+2 A c)) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx+\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a} \\ & = \frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+(c (3 b B+2 A c)) \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 )-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{8 a} \\ & = \frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {1}{2} \sqrt {c} (3 b B+2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=-\frac {\sqrt {a+x (b+c x)} \left (3 A b^2 x^2+4 a^2 (2 A+3 B x)+2 a x (3 B x (5 b-4 c x)+A (7 b+16 c x))\right )}{24 a x^3}+\frac {\left (-6 a B \left (b^2+4 a c\right )+A \left (b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {1}{2} \sqrt {c} (3 b B+2 A c) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \]
[In]
[Out]
Time = 0.57 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.23
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (32 a A c \,x^{2}+3 A \,b^{2} x^{2}+30 B a b \,x^{2}+14 a A b x +12 a^{2} B x +8 A \,a^{2}\right )}{24 x^{3} a}+\frac {16 A a \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+32 B a b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+16 B a \,c^{2} \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )-\frac {\left (12 A a b c -A \,b^{3}+24 B \,a^{2} c +6 B a \,b^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}}{16 a}\) | \(254\) |
default | \(\text {Expression too large to display}\) | \(1323\) |
[In]
[Out]
none
Time = 1.21 (sec) , antiderivative size = 953, normalized size of antiderivative = 4.63 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\left [\frac {24 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {c} x^{3} \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 ) + 3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {a} x^{3} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{2} x^{3}}, -\frac {48 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {-c} x^{3} \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 ) - 3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {a} x^{3} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 12 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {c} x^{3} \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 ) + 2 \, {\left (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 24 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {-c} x^{3} \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 (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{2} x^{3}}\right ] \]
[In]
[Out]
\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (179) = 358\).
Time = 0.35 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.04 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\sqrt {c x^{2} + b x + a} B c - \frac {{\left (3 \, B b c + 2 \, A c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} + \frac {{\left (6 \, B a b^{2} - A b^{3} + 24 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} \sqrt {c} + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} + 64 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^4} \,d x \]
[In]
[Out]