Optimal. Leaf size=1097 \[ \frac {2 (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-C (5 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)))) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^3 f^3 h^3}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {4 \sqrt {-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+C (c h (f g-e h)+d g (2 f g+e h))\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {g+h x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.14, antiderivative size = 1083, normalized size of antiderivative = 0.99, number of steps
used = 9, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1615, 1614,
1629, 164, 115, 114, 122, 121} \begin {gather*} \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a+b x)^2}{7 d f h}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac {4 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left (2 \left (6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right ) d^3+c f h \left (10 f^2 g^2+9 e f h g+10 e^2 h^2\right ) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right )\right ) b^2-7 a d f h \left (15 A d^2 f^2 h^2+C \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )\right ) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} 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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 114
Rule 115
Rule 121
Rule 122
Rule 164
Rule 1614
Rule 1615
Rule 1629
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}+\frac {\int \frac {(a+b x) \left (-4 b c C e g+7 a A d f h-a C (d e g+c f g+c e h)+(7 A b d f h-5 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h)) x+2 C (2 a d f h-3 b (d f g+d e h+c f h)) x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{7 d f h}\\ &=\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}+\frac {\int \frac {-5 a d f h (4 b c C e g-7 a A d f h+a C (d e g+c f g+c e h))-2 C (2 b c e g+a (d e g+c f g+c e h)) (2 a d f h-3 b (d f g+d e h+c f h))-2 \left (C (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)) (2 a d f h-3 b (d f g+d e h+c f h))+5 d f h \left (2 b^2 c C e g+a^2 C (d f g+d e h+c f h)-a b (7 A d f h-3 C (d e g+c f g+c e h))\right )\right ) x+(4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-5 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h))) x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{35 d^2 f^2 h^2}\\ &=\frac {2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}+\frac {2 \int \frac {\frac {1}{2} d \left (35 a^2 d^2 f^2 h^2 (3 A d f h-C (d e g+c f g+c e h))+28 a b C d f h \left (2 d^2 e g (f g+e h)+2 c^2 f h (f g+e h)+c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )-b^2 \left (35 A d^2 f^2 h^2 (d e g+c f g+c e h)+C \left (24 c^3 f^2 h^2 (f g+e h)+c^2 d f h \left (23 f^2 g^2+34 e f g h+23 e^2 h^2\right )+d^3 e g \left (24 f^2 g^2+23 e f g h+24 e^2 h^2\right )+2 c d^2 \left (12 f^3 g^3+17 e f^2 g^2 h+17 e^2 f g h^2+12 e^3 h^3\right )\right )\right )\right )-d \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{105 d^4 f^3 h^3}\\ &=\frac {2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {\left (2 \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right )\right ) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 d^3 f^3 h^4}+\frac {\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{105 d^3 f^3 h^4}\\ &=\frac {2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}+\frac {\left (\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\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}{105 d^3 f^3 h^4 \sqrt {e+f x}}-\frac {\left (2 \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) \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}{105 d^3 f^3 h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}\\ &=\frac {2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {4 \sqrt {-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) \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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {\left (\left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\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}{105 d^3 f^3 h^4 \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 \left (35 A b^2 d f h+8 a^2 C d f h-25 b^2 C (d e g+c f g+c e h)-38 a b C (d f g+d e h+c f h)+\frac {24 b^2 C (d f g+d e h+c f h)^2}{d f h}\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{105 d^2 f^2 h^2}+\frac {4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{35 d^2 f^2 h^2}+\frac {2 C (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{7 d f h}-\frac {4 \sqrt {-d e+c f} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) \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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-14 a b d f h \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c h (f g-e h)+d g (2 f g+e h))+C \left (24 c^3 f^2 h^3 (f g-e h)+c^2 d f h^2 \left (17 f^2 g^2+6 e f g h-23 e^2 h^2\right )+2 c d^2 h \left (8 f^3 g^3+e f^2 g^2 h+3 e^2 f g h^2-12 e^3 h^3\right )+d^3 g \left (48 f^3 g^3+16 e f^2 g^2 h+17 e^2 f g h^2+24 e^3 h^3\right )\right )\right )\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 )}{105 d^4 f^{7/2} h^4 \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 33.04, size = 1291, normalized size = 1.18 \begin {gather*} \frac {2 \left (-2 d^2 \sqrt {-c+\frac {d e}{f}} \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) (e+f x) (g+h x)+d^2 \sqrt {-c+\frac {d e}{f}} f h (c+d x) (e+f x) (g+h x) \left (35 a^2 C d^2 f^2 h^2-14 a b C d f h (4 c f h+d (4 f g+4 e h-3 f h x))+b^2 \left (35 A d^2 f^2 h^2+C \left (24 c^2 f^2 h^2+c d f h (23 f g+23 e h-18 f h x)+d^2 \left (24 e^2 h^2+e f h (23 g-18 h x)+3 f^2 \left (8 g^2-6 g h x+5 h^2 x^2\right )\right )\right )\right )\right )-2 i (d e-c f) h \left (35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)-7 a b d f h \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (d f g+d e h+c f h)+2 C \left (12 c^3 f^3 h^3+10 c^2 d f^2 h^2 (f g+e h)+c d^2 f h \left (10 f^2 g^2+9 e f g h+10 e^2 h^2\right )+2 d^3 \left (6 f^3 g^3+5 e f^2 g^2 h+5 e^2 f g h^2+6 e^3 h^3\right )\right )\right )\right ) (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 {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+i d h \left (35 a^2 d^2 f^2 h^2 \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right )-14 a b d f h \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (-f g+e h)+c d f \left (-4 f^2 g^2+e f g h+3 e^2 h^2\right )+d^2 e \left (4 f^2 g^2+3 e f g h+8 e^2 h^2\right )\right )\right )+b^2 \left (35 A d^2 f^2 h^2 (c f (-f g+e h)+d e (f g+2 e h))+C \left (24 c^3 f^3 h^2 (-f g+e h)+c^2 d f^2 h \left (-23 f^2 g^2+6 e f g h+17 e^2 h^2\right )+2 c d^2 f \left (-12 f^3 g^3+3 e f^2 g^2 h+e^2 f g h^2+8 e^3 h^3\right )+d^3 e \left (24 f^3 g^3+17 e f^2 g^2 h+16 e^2 f g h^2+48 e^3 h^3\right )\right )\right )\right ) (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)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )\right )}{105 d^5 \sqrt {-c+\frac {d e}{f}} f^4 h^4 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \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. \(12278\) vs.
\(2(1035)=2070\).
time = 0.12, size = 12279, normalized size = 11.19
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \,b^{2} x^{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}}{7 d f h}+\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) x \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}}{5 d f h}+\frac {2 \left (b^{2} A +C \,a^{2}-\frac {2 C \,b^{2} \left (\frac {5}{2} c e h +\frac {5}{2} c f g +\frac {5}{2} d e g \right )}{7 d f h}-\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\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}}{3 d f h}+\frac {2 \left (a^{2} A -\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) c e g}{5 d f h}-\frac {2 \left (b^{2} A +C \,a^{2}-\frac {2 C \,b^{2} \left (\frac {5}{2} c e h +\frac {5}{2} c f g +\frac {5}{2} d e g \right )}{7 d f h}-\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\right ) \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\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 )}{\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 {2 \left (2 a b A -\frac {4 C \,b^{2} c e g}{7 d f h}-\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) \left (\frac {3}{2} c e h +\frac {3}{2} c f g +\frac {3}{2} d e g \right )}{5 d f h}-\frac {2 \left (b^{2} A +C \,a^{2}-\frac {2 C \,b^{2} \left (\frac {5}{2} c e h +\frac {5}{2} c f g +\frac {5}{2} d e g \right )}{7 d f h}-\frac {2 \left (2 C a b -\frac {2 C \,b^{2} \left (3 h f c +3 d e h +3 g f d \right )}{7 d f h}\right ) \left (2 h f c +2 d e h +2 g f d \right )}{5 d f h}\right ) \left (h f c +d e h +g f d \right )}{3 d f h}\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 )}{\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}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(1238\) |
default | \(\text {Expression too large to display}\) | \(12279\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.36, size = 1769, normalized size = 1.61 \begin {gather*} \frac {2 \, {\left (3 \, {\left (15 \, C b^{2} d^{4} f^{4} h^{4} x^{2} + 24 \, C b^{2} d^{4} f^{4} g^{2} h^{2} + 24 \, C b^{2} d^{4} f^{2} h^{4} e^{2} + {\left (23 \, C b^{2} c d^{3} - 56 \, C a b d^{4}\right )} f^{4} g h^{3} + {\left (24 \, C b^{2} c^{2} d^{2} - 56 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} h^{4} - 6 \, {\left (3 \, C b^{2} d^{4} f^{4} g h^{3} + {\left (3 \, C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f^{4} h^{4}\right )} x - {\left (18 \, C b^{2} d^{4} f^{3} h^{4} x - 23 \, C b^{2} d^{4} f^{3} g h^{3} - {\left (23 \, C b^{2} c d^{3} - 56 \, C a b d^{4}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} + {\left (48 \, C b^{2} d^{4} f^{4} g^{4} + 48 \, C b^{2} d^{4} h^{4} e^{4} + 16 \, {\left (C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f^{4} g^{3} h + {\left (11 \, C b^{2} c^{2} d^{2} - 42 \, C a b c d^{3} + 70 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} g^{2} h^{2} + {\left (16 \, C b^{2} c^{3} d - 42 \, C a b c^{2} d^{2} - 210 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{4} g h^{3} + {\left (48 \, C b^{2} c^{4} - 112 \, C a b c^{3} d - 210 \, A a b c d^{3} + 315 \, A a^{2} d^{4} + 70 \, {\left (C a^{2} + A b^{2}\right )} c^{2} d^{2}\right )} f^{4} h^{4} + 16 \, {\left (C b^{2} d^{4} f g h^{3} + {\left (C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f h^{4}\right )} e^{3} + {\left (11 \, C b^{2} d^{4} f^{2} g^{2} h^{2} + 14 \, {\left (C b^{2} c d^{3} - 3 \, C a b d^{4}\right )} f^{2} g h^{3} + {\left (11 \, C b^{2} c^{2} d^{2} - 42 \, C a b c d^{3} + 70 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{2} h^{4}\right )} e^{2} + {\left (16 \, C b^{2} d^{4} f^{3} g^{3} h + 14 \, {\left (C b^{2} c d^{3} - 3 \, C a b d^{4}\right )} f^{3} g^{2} h^{2} + 7 \, {\left (2 \, C b^{2} c^{2} d^{2} - 6 \, C a b c d^{3} + 5 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{3} g h^{3} + {\left (16 \, C b^{2} c^{3} d - 42 \, C a b c^{2} d^{2} - 210 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right ) + 6 \, {\left (24 \, C b^{2} d^{4} f^{4} g^{3} h + 24 \, C b^{2} d^{4} f h^{4} e^{3} + 4 \, {\left (5 \, C b^{2} c d^{3} - 14 \, C a b d^{4}\right )} f^{4} g^{2} h^{2} + {\left (20 \, C b^{2} c^{2} d^{2} - 49 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} g h^{3} + {\left (24 \, C b^{2} c^{3} d - 56 \, C a b c^{2} d^{2} - 105 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{4} h^{4} + 4 \, {\left (5 \, C b^{2} d^{4} f^{2} g h^{3} + {\left (5 \, C b^{2} c d^{3} - 14 \, C a b d^{4}\right )} f^{2} h^{4}\right )} e^{2} + {\left (20 \, C b^{2} d^{4} f^{3} g^{2} h^{2} + {\left (18 \, C b^{2} c d^{3} - 49 \, C a b d^{4}\right )} f^{3} g h^{3} + {\left (20 \, C b^{2} c^{2} d^{2} - 49 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d f h} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right )\right )}}{315 \, d^{5} f^{5} h^{5}} \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 + C x^{2}\right ) \left (a + b x\right )^{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 (C\,x^2+A\right )\,{\left (a+b\,x\right )}^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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