Optimal. Leaf size=774 \[ -\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2-\left (c^2 \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3-\left (c^3 \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e^2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c} \]
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Rubi [A] time = 2.11, antiderivative size = 774, 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 = {941, 1653, 843, 718, 424, 419} \[ \frac {2 e \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e^2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 843
Rule 941
Rule 1653
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\int \frac {(d+e x) \left (-7 c d^2 f+e (b d f+4 a e f+a d g)-(c d (12 e f+7 d g)-e (5 b e f+2 b d g+5 a e g)) x-e (c e f+11 c d g-6 b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{7 c}\\ &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {2 \int \frac {-\frac {1}{2} g \left (6 b^2 e^3 f^2 g+b e f \left (18 a e^2 g^2-c \left (e^2 f^2+11 d e f g+5 d^2 g^2\right )\right )+c g \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )\right )-\frac {1}{2} g \left (6 b e^3 g^2 (5 b f+3 a g)-c^2 \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )-c e g \left (a e g (23 e f+63 d g)-b \left (7 e^2 f^2-85 d e f g-10 d^2 g^2\right )\right )\right ) x-\frac {1}{2} e g^2 \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{35 c^2 g^3}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {4 \int \frac {\frac {1}{4} g^3 \left (24 b^3 e^3 f g^2+b^2 e^2 g \left (24 a e g^2-c f (5 e f+84 d g)\right )-b c e \left (6 a e g^2 (11 e f+14 d g)+c f \left (4 e^2 f^2-21 d e f g-105 d^2 g^2\right )\right )-c g \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )\right )+\frac {1}{4} g^3 \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^5}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\left (e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^3}-\frac {\left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^3 g^3}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 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^4 g^3 \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 (2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 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^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{105 c^3 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 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^4 g^3 \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 {2 \sqrt {2} \sqrt {b^2-4 a c} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 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^4 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 14.67, size = 10649, normalized size = 13.76 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 14978, normalized size = 19.35 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
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 \[ \int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3} \sqrt {f + g x}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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