Optimal. Leaf size=465 \[ -\frac {2 C (a h+b g) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{b^2 d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (\frac {a^2 C}{b^2}+A\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)}+\frac {2 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \]
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Rubi [A] time = 0.81, antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ -\frac {2 \left (\frac {a^2 C}{b^2}+A\right ) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)}-\frac {2 C (a h+b g) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \]
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 1607
Rubi steps
\begin {align*} \int \frac {A+C x^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\left (A+\frac {a^2 C}{b^2}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx+\int \frac {-\frac {a C}{b^2}+\frac {C x}{b}}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\\ &=-\left (\left (2 \left (A+\frac {a^2 C}{b^2}\right )\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 )\right )+\frac {C \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{b h}-\frac {(C (b g+a h)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2 h}\\ &=-\frac {\left (2 \left (A+\frac {a^2 C}{b^2}\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 )}{\sqrt {e+f x}}-\frac {\left (C (b g+a h) \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}{b^2 h \sqrt {e+f x}}+\frac {\left (C \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}{b h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=\frac {2 C \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 d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {\left (2 \left (A+\frac {a^2 C}{b^2}\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 )}{\sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (C (b g+a h) \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}{b^2 h \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 C \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 d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 C \sqrt {-d e+c f} (b g+a h) \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 \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A+\frac {a^2 C}{b^2}\right ) \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}} \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 c-a d) \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}
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Mathematica [C] time = 8.11, size = 1036, normalized size = 2.23 \[ -\frac {2 \left (-b^2 C f \sqrt {\frac {d g}{h}-c} h c^3+b^2 C d e \sqrt {\frac {d g}{h}-c} h c^2+a b C d f \sqrt {\frac {d g}{h}-c} h c^2+2 b^2 C f \sqrt {\frac {d g}{h}-c} h (c+d x) c^2+b^2 C d f g \sqrt {\frac {d g}{h}-c} c^2-b^2 C f \sqrt {\frac {d g}{h}-c} h (c+d x)^2 c-a b C d^2 e \sqrt {\frac {d g}{h}-c} h c-b^2 C d e \sqrt {\frac {d g}{h}-c} h (c+d x) c-2 a b C d f \sqrt {\frac {d g}{h}-c} h (c+d x) c-b^2 C d f g \sqrt {\frac {d g}{h}-c} (c+d x) c-b^2 C d^2 e g \sqrt {\frac {d g}{h}-c} c-a b C d^2 f g \sqrt {\frac {d g}{h}-c} c+a b C d f \sqrt {\frac {d g}{h}-c} h (c+d x)^2+a b C d^2 e \sqrt {\frac {d g}{h}-c} h (c+d x)+a b C d^2 f g \sqrt {\frac {d g}{h}-c} (c+d x)+i b C (b c-a d) f (c h-d g) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )+i b d f (b c C g-a C d g+a c C h+A b d h) (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )-i A b^2 d^2 f h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \Pi \left (-\frac {b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )-i a^2 C d^2 f h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \Pi \left (-\frac {b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {d g}{h}-c}}{\sqrt {c+d x}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )+a b C d^3 e g \sqrt {\frac {d g}{h}-c}\right )}{b^2 d^2 (b c-a d) f \sqrt {\frac {d g}{h}-c} h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1368, normalized size = 2.94 \[ \text {result too large to display} \]
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 \[ \int \frac {C x^{2} + A}{{\left (b x + a\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
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 \[ \int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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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