Optimal. Leaf size=652 \[ -\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right )}{105 c^2 g^2}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\left (-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right ) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 1.11, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\left (-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right ) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right )}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 814
Rule 832
Rule 843
Rubi steps
\begin {align*} \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx &=\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} (7 c d f-3 b e f-a e g)+\frac {1}{2} (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx}{7 c}\\ &=-\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {4 \int \frac {\frac {1}{4} \left (5 c g (b f-2 a g) (7 c d f-e (3 b f+a g))-2 (c e f+7 c d g-4 b e g) \left (\frac {1}{2} b f (4 c f-b g)-a g \left (c f+\frac {b g}{2}\right )\right )\right )-\frac {1}{4} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^2}\\ &=-\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (\left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^3}+\frac {\left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^3}\\ &=-\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a 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 )}{105 c^3 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} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\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^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a 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 )}{105 c^3 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} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\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^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 13.95, size = 8432, normalized size = 12.93 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \sqrt {g x + f}, x\right ) \]
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 \[ \int \sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \sqrt {g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 10711, normalized size = 16.43 \[ \text {output too large to display} \]
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 \[ \int \sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \sqrt {g x + f}\,{d x} \]
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 \[ \int \sqrt {f+g\,x}\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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 \[ \int \left (d + e x\right ) \sqrt {f + g x} \sqrt {a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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