Optimal. Leaf size=755 \[ -\frac {4 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2-\left (c^2 \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 e (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g} \]
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Rubi [A] time = 1.93, antiderivative size = 755, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {920, 1653, 843, 718, 424, 419} \[ -\frac {4 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (-\left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )\right )}{105 c^2 g^3}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (35 d^2 g^2-42 d e f g+16 e^2 f^2\right )\right )+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 e (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 843
Rule 920
Rule 1653
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx &=\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {\int \frac {(d+e x) \left (b d f+4 a e f-6 a d g+(2 c d f+5 b e f-5 b d g-2 a e g) x+(6 c e f-4 c d g-b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{7 g}\\ &=\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^2 f^2 g-2 a c g \left (9 e^2 f^2-16 d e f g+15 d^2 g^2\right )+b f \left (3 a e^2 g^2-c \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )\right )\right )+\frac {1}{2} g \left (b e^2 g^2 (5 b f+3 a g)-2 c^2 f \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )+c g \left (2 a e g (e f-14 d g)-b \left (28 e^2 f^2-50 d e f g+25 d^2 g^2\right )\right )\right ) x+g^2 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{35 c g^4}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}-\frac {4 \int \frac {-\frac {1}{4} g^3 \left (4 b^3 e^2 f g^2+b^2 e g \left (4 a e g^2+c f (5 e f-14 d g)\right )-b c \left (a e g^2 (11 e f+14 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )-2 a c g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right )\right )-\frac {1}{4} g^3 \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^6}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (2 \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4}+\frac {\left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^4}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \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} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {4 \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)-c^2 \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^2 g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 g}-\frac {2 e (6 c e f-4 c d g-b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c g^3}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^3 e^2 g^3+b c e g^2 (9 b e f-28 b d g-29 a e g)-2 c^3 f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-c^2 g \left (2 a e g (13 e f-42 d g)-b \left (16 e^2 f^2-42 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \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} \left (c f^2-b f g+a g^2\right ) \left (2 b^2 e^2 g^2+c e g (4 b e f-7 b d g-5 a e g)+c^2 \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\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 )}{105 c^3 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 14.14, size = 10030, normalized size = 13.28 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {c x^{2} + b x + a}}{\sqrt {g x + f}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 12923, normalized size = 17.12 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}} \,d x \]
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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