Integrand size = 37, antiderivative size = 968 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{d h \sqrt {e+f x}}-\frac {b \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {b (d e-c f) (b f g+b e h-2 a f h) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{d f^2 h \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {b \sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{d f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {b c-a d} \sqrt {-d g+c h} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d h \sqrt {c+d x} \sqrt {e+f x}} \]
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Time = 0.61 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {172, 179, 182, 435, 171, 551, 176, 430} \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {\sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\arcsin \left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right ) b}{d f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {(d e-c f) (b f g+b e h-2 a f h) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) b}{d f^2 h \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right ) b}{d f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x} b}{d h \sqrt {e+f x}}-\frac {2 \sqrt {b c-a d} \sqrt {c h-d g} (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \operatorname {EllipticPi}\left (-\frac {b (d g-c h)}{(b c-a d) h},\arcsin \left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {c h-d g} \sqrt {a+b x}}\right ),\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d h \sqrt {c+d x} \sqrt {e+f x}} \]
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Rule 171
Rule 172
Rule 176
Rule 179
Rule 182
Rule 430
Rule 435
Rule 551
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{d} \\ & = \frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{d h \sqrt {e+f x}}-\frac {(b (d e-c f) (f g-e h)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} (e+f x)^{3/2} \sqrt {g+h x}} \, dx}{2 d f h}+\frac {(b (d e-c f) (b f g+b e h-2 a f h)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 d f^2 h}+\frac {(b (a d f h-b (d f g+d e h-c f h))) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}} \, dx}{2 d f^2 h}-\frac {\left (2 (b c-a d) (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 )}{d \sqrt {c+d x} \sqrt {e+f x}} \\ & = \frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{d h \sqrt {e+f x}}-\frac {2 \sqrt {b c-a d} \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 )}{d h \sqrt {c+d x} \sqrt {e+f x}}+\frac {\left (b (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\left (h-f x^2\right ) \sqrt {1+\frac {(-b e+a f) x^2}{b g-a h}} \sqrt {1+\frac {(-d e+c f) x^2}{d g-c h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {e+f x}}\right )}{d f^2 h \sqrt {a+b x} \sqrt {c+d x}}+\frac {\left (b (d e-c f) (b f g+b e h-2 a f h) \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 )}{d f^2 h (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 {\left (b (d e-c f) (f g-e h) \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {(-b e+a f) x^2}{b c-a d}}}{\sqrt {1-\frac {(f g-e h) x^2}{d g-c h}}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{d f (-d e+c f) h \sqrt {\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}} \\ & = \frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{d h \sqrt {e+f x}}-\frac {b \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {\frac {(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d f h \sqrt {-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt {g+h x}}+\frac {b (d e-c f) (b f g+b e h-2 a f h) \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 )}{d f^2 h \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 {b \sqrt {b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac {f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{d f^2 \sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {b c-a d} \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 )}{d h \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(7319\) vs. \(2(968)=1936\).
Time = 30.31 (sec) , antiderivative size = 7319, normalized size of antiderivative = 7.56 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
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Time = 2.60 (sec) , antiderivative size = 1541, normalized size of antiderivative = 1.59
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1541\) |
default | \(\text {Expression too large to display}\) | \(17031\) |
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
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\[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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\[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
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