Optimal. Leaf size=451 \[ -\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )+8 \left (2 a^2 f^2 (B f+2 C e)-b^2 e \left (C e^2-2 f (5 A f+B e)\right )\right )\right )}{120 b^4 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^4}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x}}{6 b^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.00967, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )+8 \left (2 a^2 f^2 (B f+2 C e)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (5 A f+B e)\right )\right )\right )}{120 b^4 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)+a^4 C f^2\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)+a^4 C f^2\right )}{16 b^4}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x}}{6 b^2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
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)^2 \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)^2 \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)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac{\left (\sqrt{a+b x} \sqrt{a c-b c x}\right ) \int (e+f x)^2 \left (-3 c \left (2 A b^2+a^2 C\right ) f^2+3 b^2 c f (C e-2 B f) x\right ) \sqrt{a^2 c-b^2 c x^2} \, dx}{6 b^2 c f^2 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{(C e-2 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 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^2 f^2 \left (10 A b^2 e+a^2 (3 C e+4 B f)\right )+3 b^2 c^2 f \left (5 \left (2 A b^2+a^2 C\right ) f^2-2 b^2 e (C e-2 B f)\right ) x\right ) \sqrt{a^2 c-b^2 c x^2} \, dx}{30 b^4 c^2 f^2 \sqrt{a^2 c-b^2 c x^2}}\\ &=\frac{(C e-2 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac{\left (\left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}+\frac{(C e-2 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac{\left (a^2 c \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}+\frac{(C e-2 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac{\left (a^2 c \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt{a+b x} \sqrt{a c-b c x}}{16 b^4}+\frac{(C e-2 B f) \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac{C \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac{\sqrt{a+b x} \sqrt{a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac{a^2 \sqrt{c} \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 = 0.995131, size = 311, normalized size = 0.69 \[ \frac{\sqrt{c (a-b x)} \left (b \left (a^2-b^2 x^2\right ) \left (2 a^2 b^2 \left (5 A f (16 e+3 f x)+B \left (40 e^2+30 e f x+8 f^2 x^2\right )+C x \left (15 e^2+16 e f x+5 f^2 x^2\right )\right )+a^4 f (32 B f+64 C e+15 C f x)-4 b^4 x \left (5 A \left (6 e^2+8 e f x+3 f^2 x^2\right )+x \left (2 B \left (10 e^2+15 e f x+6 f^2 x^2\right )+C x \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )\right )+30 a^{5/2} \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right ) \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)+a^4 C f^2\right )\right )}{240 b^5 (b x-a) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 987, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.30742, size = 1517, normalized size = 3.36 \begin{align*} \left [\frac{15 \,{\left (4 \, B a^{4} b^{2} e f + 2 \,{\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} +{\left (C a^{6} + 2 \, A a^{4} b^{2}\right )} f^{2}\right )} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \,{\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \,{\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f -{\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \,{\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \,{\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \,{\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \,{\left (4 \, B a^{2} b^{3} e f + 2 \,{\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} +{\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{480 \, b^{5}}, -\frac{15 \,{\left (4 \, B a^{4} b^{2} e f + 2 \,{\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} +{\left (C a^{6} + 2 \, A a^{4} b^{2}\right )} f^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{c} x}{b^{2} c x^{2} - a^{2} c}\right ) -{\left (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \,{\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \,{\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f -{\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \,{\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \,{\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \,{\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \,{\left (4 \, B a^{2} b^{3} e f + 2 \,{\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} +{\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{240 \, b^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- c \left (- a + b x\right )} \sqrt{a + b x} \left (e + f x\right )^{2} \left (A + B x + C x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]