Optimal. Leaf size=981 \[ \frac {(5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {(b e-a f) \sqrt {b g-a h} (3 a B d f h+b (4 A d f h-B (c f h+3 d (f g+e h)))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b d f^2 h^2 \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac {\sqrt {-d g+c h} (4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))-(a d f h+b (d f g+d e h+c f h)) (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h)))) (a+b x) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac {b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {g+h x}}{\sqrt {-d g+c h} \sqrt {a+b x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}} \]
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Rubi [A]
time = 2.13, antiderivative size = 976, normalized size of antiderivative = 0.99, number of steps
used = 10, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1611, 1614,
1616, 1612, 176, 430, 171, 551, 182, 435} \begin {gather*} \frac {b \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} B}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\text {ArcSin}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {(b e-a f) \sqrt {b g-a h} (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e h))) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\text {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 )}{4 b d f^2 h^2 \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 {c h-d g} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f 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};\text {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 )}{4 b d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}}+\frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 171
Rule 176
Rule 182
Rule 430
Rule 435
Rule 551
Rule 1611
Rule 1612
Rule 1614
Rule 1616
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {\int \frac {\sqrt {a+b x} \left (6 a A d f h+6 (A b+a B) d f h x+6 b B d f h x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 d f h}\\ &=\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {6 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )+12 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+6 b d f h (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{24 d^2 f^2 h^2}\\ &=\frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}+\frac {\int \frac {-6 b d f h \left ((b d e g+a c f h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-2 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )\right )-6 b d f h ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{48 b d^3 f^3 h^3}+\frac {((d e-c f) (d g-c h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 d^2 f^2 h^2}\\ &=\frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e h)))) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d f^2 h^2}-\frac {((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{8 b d^2 f^2 h^2}-\frac {\left ((d g-c h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {(-b c+a d) x^2}{b e-a f}}}{\sqrt {1-\frac {(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {c+d x}}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}\\ &=\frac {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left (((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f 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)}}\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 )}{4 b d^2 f^2 h^2 \sqrt {c+d x} \sqrt {e+f x}}-\frac {\left ((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e 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 )}{4 b d f^2 h^2 (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 {(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{4 d f^2 h^2 \sqrt {c+d x}}+\frac {b B \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{2 d f h}-\frac {\sqrt {d g-c h} \sqrt {f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {(b e-a f) \sqrt {b g-a h} (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e 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 )}{4 b d f^2 h^2 \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 {-d g+c h} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f 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 )}{4 b d^2 \sqrt {b c-a d} f^2 h^3 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(21611\) vs. \(2(981)=1962\).
time = 35.79, size = 21611, normalized size = 22.03 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55942\) vs.
\(2(898)=1796\).
time = 0.12, size = 55943, normalized size = 57.03
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1814\) |
default | \(\text {Expression too large to display}\) | \(55943\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \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 {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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