Optimal. Leaf size=738 \[ \frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac {\sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (b e-a f) (b g-a h)} \]
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Rubi [A] time = 1.91, antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1605, 1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d) (b e-a f)}-\frac {\sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a^3 b C (c f h+d e h+d f g)+a^4 C d f h-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}+\frac {\sqrt {f} \sqrt {g+h x} \left (\frac {a^2 C}{b}+A b\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {d (g+h x)}{d g-c h}}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 120
Rule 121
Rule 158
Rule 169
Rule 537
Rule 538
Rule 1605
Rule 1607
Rubi steps
\begin {align*} \int \frac {A+C x^2}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=-\frac {\left (A b^2+a^2 C\right ) \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 b^2 (d e g+c f g+c e h)-2 a b (c C e g-A d f g-A d e h-A c f h)-a^2 (2 A d f h-C (d e g+c f g+c e h))+2 \left (b^2 c C e g+a^2 C (d f g+d e h+c f h)+a b (A d f h-C (d e g+c f g+c e h))\right ) x+\left (A b^2+a^2 C\right ) 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 {\left (A b^2+a^2 C\right ) \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 b c C e g-2 a C d e g-2 a c C f g+\frac {2 a^2 C d f g}{b}-2 a c C e h+\frac {2 a^2 C d e h}{b}+\frac {2 a^2 c C f h}{b}+a A d f h-\frac {a^3 C d f h}{b^2}+\left (A b d f h+\frac {a^2 C d f h}{b}\right ) 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 (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {\left (A b^2+a^2 C\right ) \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^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{2 b^2 (b c-a d) (b e-a f)}+\frac {\left (\left (A b+\frac {a^2 C}{b}\right ) d f\right ) \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 (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \operatorname {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^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {\left (A b^2+a^2 C\right ) \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 (\left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \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^2 (b c-a d) (b e-a f) \sqrt {e+f x}}-\frac {\left (\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \operatorname {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^2 (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x}}+\frac {\left (\left (A b+\frac {a^2 C}{b}\right ) 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 {\left (A b^2+a^2 C\right ) \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+\frac {a^2 C}{b}\right ) \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 (\left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \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^2 (b c-a d) (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (\left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \operatorname {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^2 (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x} \sqrt {g+h x}}\\ &=-\frac {\left (A b^2+a^2 C\right ) \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+\frac {a^2 C}{b}\right ) \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 {\sqrt {-d e+c f} \left (a^2 C d f-2 a b C (d e+c f)+b^2 (2 c C e-A d f)\right ) \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^2 d (b c-a d) \sqrt {f} (b e-a f) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\sqrt {-d e+c f} \left (a^4 C d f h-A b^4 (d e g+c f g+c e h)-2 a^3 b C (d f g+d e h+c f h)-2 a b^3 (2 c C e g-A d f g-A d e h-A c f h)-3 a^2 b^2 (A d f h-C (d e g+c f g+c e 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^2 (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] time = 15.20, size = 637, normalized size = 0.86 \[ \frac {b^2 d (c+d x) (e+f x) (g+h x) \left (a^2 C+A b^2\right ) (a d-b c) \sqrt {\frac {d g}{h}-c}+(a+b x) \left (b d^2 (e+f x) (g+h x) \left (a^2 C+A b^2\right ) (b c-a d) \sqrt {\frac {d g}{h}-c}-i (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (-b (b e-a f) \left (a^2 C d (c h-d g)-2 a b h \left (A d^2+c^2 C\right )+b^2 \left (A c d h+A d^2 g+2 c^2 C g\right )\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )+b f \left (a^2 C+A b^2\right ) (b c-a d) (c h-d g) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )-d \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)+3 a^2 b^2 (C (c e h+c f g+d e g)-A d f h)+2 a b^3 (A c f h+A d e h+A d f g-2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac {b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )\right )\right )}{b^2 d (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2 (b e-a f) (b g-a h) \sqrt {\frac {d g}{h}-c}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 17460, normalized size = 23.66 \[ \text {output too large to display} \]
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 \[ \int \frac {C x^{2} + A}{{\left (b x + a\right )}^{2} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
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 \[ \int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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