Optimal. Leaf size=519 \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]
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Rubi [A] time = 0.54, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(d+e x) \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx &=-\frac {2 \sqrt {f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt {a+b x+c x^2}}{15 c g^2}-\frac {2 \int \frac {\frac {1}{2} \left (5 c d g (b f-2 a g)-b e f (4 c f-b g)+2 a e g \left (c f+\frac {b g}{2}\right )\right )+\frac {1}{2} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c g^2}\\ &=-\frac {2 \sqrt {f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt {a+b x+c x^2}}{15 c g^2}-\frac {\left ((8 c e f-10 c d g+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 g^3}-\frac {\left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c g^3}\\ &=-\frac {2 \sqrt {f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt {a+b x+c x^2}}{15 c g^2}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\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^2 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} (8 c e f-10 c d g+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^2 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {f+g x} (4 c e f-5 c d g-b e g-3 c e g x) \sqrt {a+b x+c x^2}}{15 c g^2}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 e g^2-2 c^2 f (4 e f-5 d g)+c g (3 b e f-5 b d g-6 a e g)\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^2 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} (8 c e f-10 c d g+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^2 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 10.51, size = 911, normalized size = 1.76 \[ \frac {2 \sqrt {a+x (b+c x)} \left (\left (2 f (4 e f-5 d g) c^2+g (-3 b e f+5 b d g+6 a e g) c-2 b^2 e g^2\right ) \left (c \left (\frac {f}{f+g x}-1\right )^2+\frac {g \left (-\frac {f b}{f+g x}+b+\frac {a g}{f+g x}\right )}{f+g x}\right )+\frac {i \sqrt {1-\frac {2 \left (c f^2+g (a g-b f)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 \left (c f^2+g (a g-b f)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}+1} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (2 f (5 d g-4 e f) c^2+g (3 b e f-5 b d g-6 a e g) c+2 b^2 e g^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+\left (2 b^3 e g^3-b^2 \left (-c f e+2 \sqrt {\left (b^2-4 a c\right ) g^2} e+5 c d g\right ) g^2+b c \left (\sqrt {\left (b^2-4 a c\right ) g^2} (5 d g-3 e f)-8 a e g^2\right ) g+2 c \left (a \left (-2 c f e+3 \sqrt {\left (b^2-4 a c\right ) g^2} e+10 c d g\right ) g^2+c f \sqrt {\left (b^2-4 a c\right ) g^2} (4 e f-5 d g)\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c f^2+g (a g-b f)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right ) (f+g x)^{3/2}}{15 c^2 g^4 \sqrt {c x^2+b x+a} \sqrt {\frac {(f+g x)^2 \left (c \left (\frac {f}{f+g x}-1\right )^2+\frac {g \left (-\frac {f b}{f+g x}+b+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}}+\left (\frac {2 (-4 c e f+5 c d g+b e g)}{15 c g^2}+\frac {2 e x}{5 g}\right ) \sqrt {a+x (b+c x)} \sqrt {f+g x} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}{\sqrt {g x + f}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}{\sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 6207, normalized size = 11.96 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}}{\sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}{\sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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