Optimal. Leaf size=678 \[ -\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}+\frac {\sqrt {-d e+c f} \left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}} \]
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Rubi [A]
time = 1.01, antiderivative size = 678, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1613, 1621,
175, 552, 551, 164, 115, 114, 122, 121} \begin {gather*} \frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac {\sqrt {f} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {e+f x} \sqrt {g+h x} (b c-a d) (b e-a f)}+\frac {\sqrt {f} \sqrt {g+h x} (A b-a B) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)} \end {gather*}
Antiderivative was successfully verified.
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Rule 114
Rule 115
Rule 121
Rule 122
Rule 164
Rule 175
Rule 551
Rule 552
Rule 1613
Rule 1621
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {\int \frac {-2 a^2 A d f h+b^2 (2 B c e g-A (d e g+c f g+c e h))-a b (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))+2 a (A b-a B) d f h x+b (A b-a B) d f h x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {\int \frac {a A d f h-\frac {a^2 B d f h}{b}+(A b d f h-a B d f h) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac {\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac {((A b-a B) d f) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b (b c-a d) (b e-a f)}+\frac {((A b-a B) d f) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}+\frac {\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac {\left ((A b-a B) d f \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{2 b (b c-a d) (b e-a f) \sqrt {e+f x}}+\frac {\left (\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x}}+\frac {\left ((A b-a B) d f \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left ((A b-a B) d f \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{2 b (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}+\frac {\left (\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {1+\frac {h x^2}{d \left (g-\frac {c h}{d}\right )}}} \, dx,x,\sqrt {c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}}\\ &=-\frac {b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(A b-a B) \sqrt {f} \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}+\frac {\sqrt {-d e+c f} \left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d)^2 \sqrt {f} (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 33.97, size = 3402, normalized size = 5.02 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(13343\) vs.
\(2(621)=1242\).
time = 0.12, size = 13344, normalized size = 19.68
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {b \left (A b -B a \right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{\left (a^{3} d f h -a^{2} b c f h -a^{2} b d e h -a^{2} b d f g +a \,b^{2} c e h +a \,b^{2} c f g +a \,b^{2} d e g -b^{3} c e g \right ) \left (b x +a \right )}-\frac {a d f h \left (A b -B a \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\left (a^{3} d f h -a^{2} b c f h -a^{2} b d e h -a^{2} b d f g +a \,b^{2} c e h +a \,b^{2} c f g +a \,b^{2} d e g -b^{3} c e g \right ) b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}-\frac {d f h \left (A b -B a \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\left (a^{3} d f h -a^{2} b c f h -a^{2} b d e h -a^{2} b d f g +a \,b^{2} c e h +a \,b^{2} c f g +a \,b^{2} d e g -b^{3} c e g \right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {\left (3 a^{2} A b d f h -2 A a \,b^{2} c f h -2 A a \,b^{2} d e h -2 A a \,b^{2} d f g +A \,b^{3} c e h +A \,b^{3} c f g +A \,b^{3} d e g -a^{3} B d f h +B a \,b^{2} c e h +B a \,b^{2} c f g +B a \,b^{2} d e g -2 B \,b^{3} c e g \right ) \left (-\frac {e}{f}+\frac {g}{h}\right ) \sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {g}{h}}{-\frac {e}{f}+\frac {g}{h}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\left (a^{3} d f h -a^{2} b c f h -a^{2} b d e h -a^{2} b d f g +a \,b^{2} c e h +a \,b^{2} c f g +a \,b^{2} d e g -b^{3} c e g \right ) b^{2} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(1208\) |
default | \(\text {Expression too large to display}\) | \(13344\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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