Optimal. Leaf size=236 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.241952, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1661, 640, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1661
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (6 c d-a f+\frac{1}{2} (12 c e-7 b f) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (2 c (6 c d-a f)-\frac{1}{2} b (12 c e-7 b f)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.696889, size = 392, normalized size = 1.66 \[ \frac{\frac{360 d \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{3/2}}-60 b e \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )+\frac{f \left (5 \left (7 b^2-4 a c\right ) \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )-1792 b (a+x (b+c x))^{5/2}\right )}{c}+1920 d (b+2 c x) (a+x (b+c x))^{3/2}+3072 e (a+x (b+c x))^{5/2}+2560 f x (a+x (b+c x))^{5/2}}{15360 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.053, size = 862, normalized size = 3.7 \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 = 2.12967, size = 1947, normalized size = 8.25 \begin{align*} \text{result too large to display} \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 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19662, size = 563, normalized size = 2.39 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c f x + \frac{13 \, b c^{5} f + 12 \, c^{6} e}{c^{5}}\right )} x + \frac{120 \, c^{6} d + 3 \, b^{2} c^{4} f + 140 \, a c^{5} f + 132 \, b c^{5} e}{c^{5}}\right )} x + \frac{360 \, b c^{5} d - 7 \, b^{3} c^{3} f + 36 \, a b c^{4} f + 12 \, b^{2} c^{4} e + 384 \, a c^{5} e}{c^{5}}\right )} x + \frac{120 \, b^{2} c^{4} d + 2400 \, a c^{5} d + 35 \, b^{4} c^{2} f - 216 \, a b^{2} c^{3} f + 240 \, a^{2} c^{4} f - 60 \, b^{3} c^{3} e + 336 \, a b c^{4} e}{c^{5}}\right )} x - \frac{360 \, b^{3} c^{3} d - 2400 \, a b c^{4} d + 105 \, b^{5} c f - 760 \, a b^{3} c^{2} f + 1296 \, a^{2} b c^{3} f - 180 \, b^{4} c^{2} e + 1200 \, a b^{2} c^{3} e - 1536 \, a^{2} c^{4} e}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d - 192 \, a b^{2} c^{3} d + 384 \, a^{2} c^{4} d + 7 \, b^{6} f - 60 \, a b^{4} c f + 144 \, a^{2} b^{2} c^{2} f - 64 \, a^{3} c^{3} f - 12 \, b^{5} c e + 96 \, a b^{3} c^{2} e - 192 \, a^{2} b c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]