Optimal. Leaf size=591 \[ -\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{70 b^4 f}-\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (3 b^2 f x \left (a^2 f^2 (35 B f+41 C e)-2 b^2 e \left (3 C e^2-7 f (7 A f+B e)\right )\right )+8 \left (2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )+8 a^4 C f^4+b^4 \left (-e^2\right ) \left (3 C e^2-7 f (12 A f+B e)\right )\right )\right )}{840 b^6 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+a^2 \left (a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right )\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+a^2 \left (a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right )\right )}{16 b^4}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x} (3 C e-7 B f)}{42 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^4 \sqrt{a c-b c x}}{7 b^2 f} \]
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Rubi [A] time = 1.5174, antiderivative size = 584, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1610, 1654, 833, 780, 195, 217, 203} \[ \frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} \left (-\frac{8 a^2 C f^2}{b^2}-7 f (2 A f+B e)+3 C e^2\right )}{70 b^2 f}-\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (3 b^2 f x \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )+8 \left (2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )+8 a^4 C f^4+b^4 \left (-\left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )\right )}{840 b^6 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)+a^4 f^2 (B f+3 C e)\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)+a^4 f^2 (B f+3 C e)\right )}{16 b^4}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x} (3 C e-7 B f)}{42 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^4 \sqrt{a c-b c x}}{7 b^2 f} \]
Antiderivative was successfully verified.
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Rule 1610
Rule 1654
Rule 833
Rule 780
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx &=\frac{\left (\sqrt{a+b x} \sqrt{a c-b c x}\right ) \int (e+f x)^3 \sqrt{a^2 c-b^2 c x^2} \left (A+B x+C x^2\right ) \, dx}{\sqrt{a^2 c-b^2 c x^2}}\\ &=-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\left (\sqrt{a+b x} \sqrt{a c-b c x}\right ) \int (e+f x)^3 \left (-c \left (7 A b^2+4 a^2 C\right ) f^2+b^2 c f (3 C e-7 B f) x\right ) \sqrt{a^2 c-b^2 c x^2} \, dx}{7 b^2 c f^2 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}+\frac{\left (\sqrt{a+b x} \sqrt{a c-b c x}\right ) \int (e+f x)^2 \left (3 b^2 c^2 f^2 \left (14 A b^2 e+a^2 (5 C e+7 B f)\right )+3 b^2 c^2 f \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) x\right ) \sqrt{a^2 c-b^2 c x^2} \, dx}{42 b^4 c^2 f^2 \sqrt{a^2 c-b^2 c x^2}}\\ &=-\frac{\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\left (\sqrt{a+b x} \sqrt{a c-b c x}\right ) \int (e+f x) \left (-3 b^2 c^3 f^2 \left (16 a^4 C f^2+a^2 b^2 e (19 C e+49 B f)+14 A \left (5 b^4 e^2+2 a^2 b^2 f^2\right )\right )-3 b^4 c^3 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \sqrt{a^2 c-b^2 c x^2} \, dx}{210 b^6 c^3 f^2 \sqrt{a^2 c-b^2 c x^2}}\\ &=-\frac{\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac{\left (\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x}\right ) \int \sqrt{a^2 c-b^2 c x^2} \, dx}{8 b^4 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}-\frac{\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac{\left (a^2 c \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x}\right ) \int \frac{1}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{16 b^4 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}-\frac{\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac{\left (a^2 c \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1+b^2 c x^2} \, dx,x,\frac{x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{16 b^4 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}-\frac{\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac{(3 C e-7 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac{a^2 \sqrt{c} \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.40802, size = 427, normalized size = 0.72 \[ \frac{\sqrt{c (a-b x)} \left (\left (a^2-b^2 x^2\right ) \left (a^4 b^2 f \left (7 f (32 A f+96 B e+15 B f x)+C \left (672 e^2+315 e f x+64 f^2 x^2\right )\right )+2 a^2 b^4 \left (7 A f \left (120 e^2+45 e f x+8 f^2 x^2\right )+7 B \left (45 e^2 f x+40 e^3+24 e f^2 x^2+5 f^3 x^3\right )+3 C x \left (56 e^2 f x+35 e^3+35 e f^2 x^2+8 f^3 x^3\right )\right )+128 a^6 C f^3-4 b^6 x \left (21 A \left (20 e^2 f x+10 e^3+15 e f^2 x^2+4 f^3 x^3\right )+x \left (7 B \left (45 e^2 f x+20 e^3+36 e f^2 x^2+10 f^3 x^3\right )+3 C x \left (84 e^2 f x+35 e^3+70 e f^2 x^2+20 f^3 x^3\right )\right )\right )\right )+210 a^{5/2} b \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right ) \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)+a^4 f^2 (B f+3 C e)\right )\right )}{1680 b^6 (b x-a) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 1446, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45001, size = 2147, normalized size = 3.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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