Integrand size = 62, antiderivative size = 436 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b B-2 a C) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{\sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \]
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Time = 0.37 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {24, 1612, 176, 430, 171, 551} \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {g+h x} (b B-2 a C) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 C (a+b x) \sqrt {c h-d g} \sqrt {\frac {(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt {\frac {(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{h \sqrt {c+d x} \sqrt {e+f x} \sqrt {b c-a d}} \]
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Rule 24
Rule 171
Rule 176
Rule 430
Rule 551
Rule 1612
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b^2 (b B-a C)+b^3 C x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2} \\ & = C \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx+(b B-2 a C) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx \\ & = \frac {\left (2 C (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}\right ) \text {Subst}\left (\int \frac {1}{\left (h-b x^2\right ) \sqrt {1+\frac {(b c-a d) x^2}{d g-c h}} \sqrt {1+\frac {(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {a+b x}}\right )}{\sqrt {c+d x} \sqrt {e+f x}}+\frac {\left (2 (b B-2 a C) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}} \sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{(f g-e h) \sqrt {c+d x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}} \\ & = \frac {2 (b B-2 a C) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {2 C \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{\sqrt {b c-a d} h \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}
Time = 25.18 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.34 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (a+b x)^{3/2} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \left (-\frac {b B \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b g-a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 a C \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} (g+h x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(-b g+a h) (a+b x) \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {C (-f g+e h) \sqrt {-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \operatorname {EllipticPi}\left (\frac {b (-f g+e h)}{(b e-a f) h},\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(-b c+a d) (-f g+e h)}{(b e-a f) (d g-c h)}\right )}{(b e-a f) h}\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(855\) vs. \(2(398)=796\).
Time = 6.31 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.96
method | result | size |
elliptic | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 \left (B b -C a \right ) \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}+\frac {2 C b \left (\frac {g}{h}-\frac {a}{b}\right ) \sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (x +\frac {c}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {e}{f}\right )}{\left (-\frac {e}{f}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c}{d}+\frac {a}{b}\right ) \left (x +\frac {g}{h}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}\, \left (-\frac {c F\left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )}{d}+\left (\frac {c}{d}-\frac {a}{b}\right ) \Pi \left (\sqrt {\frac {\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (x +\frac {a}{b}\right )}{\left (-\frac {g}{h}+\frac {a}{b}\right ) \left (x +\frac {c}{d}\right )}}, \frac {-\frac {g}{h}+\frac {a}{b}}{-\frac {g}{h}+\frac {c}{d}}, \sqrt {\frac {\left (\frac {e}{f}-\frac {c}{d}\right ) \left (\frac {g}{h}-\frac {a}{b}\right )}{\left (-\frac {a}{b}+\frac {e}{f}\right ) \left (-\frac {c}{d}+\frac {g}{h}\right )}}\right )\right )}{\left (-\frac {g}{h}+\frac {c}{d}\right ) \left (-\frac {c}{d}+\frac {a}{b}\right ) \sqrt {b d f h \left (x +\frac {a}{b}\right ) \left (x +\frac {c}{d}\right ) \left (x +\frac {e}{f}\right ) \left (x +\frac {g}{h}\right )}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(856\) |
default | \(\text {Expression too large to display}\) | \(2955\) |
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Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {B b - C a + C b x}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
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\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]
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