Optimal. Leaf size=567 \[ \frac {4 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} (2 b e g-5 c d g+c e f) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e g (9 a e g+20 b d g+3 b e f)+8 b^2 e^2 g^2-\left (c^2 \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} (-4 b e g+7 c d g+c e f)}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.93, antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {941, 1653, 843, 718, 424, 419} \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e g (9 a e g+20 b d g+3 b e f)+8 b^2 e^2 g^2+c^2 \left (-\left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} (2 b e g-5 c d g+c e f) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} (-4 b e g+7 c d g+c e f)}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 424
Rule 718
Rule 843
Rule 941
Rule 1653
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\int \frac {-5 c d^2 f+e (b d f+2 a e f+a d g)-(c d (8 e f+5 d g)-e (3 b e f+2 b d g+3 a e g)) x-e (c e f+7 c d g-4 b e g) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {2 \int \frac {-\frac {1}{2} g \left (4 b^2 e^2 f g+b e \left (4 a e g^2-c f (e f+10 d g)\right )+c g \left (15 c d^2 f-a e (7 e f+10 d g)\right )\right )-\frac {1}{2} g \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (2 e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}+\frac {\left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 c^3 g^2 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 11.44, size = 1002, normalized size = 1.77 \[ \frac {\left (\frac {2 e^2 x}{5 c}-\frac {2 e (-c e f-10 c d g+4 b e g)}{15 c^2 g}\right ) \sqrt {f+g x} \left (c x^2+b x+a\right )}{\sqrt {a+x (b+c x)}}-\frac {2 (f+g x)^{3/2} \sqrt {c x^2+b x+a} \left (\left (\left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right ) c^2+e g (3 b e f+20 b d g+9 a e g) c-8 b^2 e^2 g^2\right ) \left (c \left (\frac {f}{f+g x}-1\right )^2+\frac {g \left (-\frac {f b}{f+g x}+b+\frac {a g}{f+g x}\right )}{f+g x}\right )+\frac {i \sqrt {1-\frac {2 \left (c f^2+g (a g-b f)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 \left (c f^2+g (a g-b f)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}+1} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (\left (-2 e^2 f^2+10 d e g f+15 d^2 g^2\right ) c^2-e g (3 b e f+20 b d g+9 a e g) c+8 b^2 e^2 g^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+\left (-30 d^2 f g^2 c^3-\left (-15 b d (2 e f+d g) g^2-2 a e (7 e f+10 d g) g^2+\sqrt {\left (b^2-4 a c\right ) g^2} \left (-2 e^2 f^2+10 d e g f+15 d^2 g^2\right )\right ) c^2+e g \left (-g (11 e f+20 d g) b^2-17 a e g^2 b+\sqrt {\left (b^2-4 a c\right ) g^2} (3 e f+20 d g) b+9 a e g \sqrt {\left (b^2-4 a c\right ) g^2}\right ) c+8 b^2 e^2 g^2 \left (b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c f^2+g (a g-b f)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right )}{15 c^3 g^3 \sqrt {a+x (b+c x)} \sqrt {\frac {(f+g x)^2 \left (c \left (\frac {f}{f+g x}-1\right )^2+\frac {g \left (-\frac {f b}{f+g x}+b+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 8248, normalized size = 14.55 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{2} \sqrt {f + g x}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________