Optimal. Leaf size=510 \[ -\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac{3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.765908, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {744, 834, 806, 720, 724, 206} \[ -\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac{3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 744
Rule 834
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx &=-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{\int \frac{\left (\frac{1}{2} (-14 c d+9 b e)+2 c e x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx}{7 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}+\frac{\int \frac{\left (\frac{3}{4} \left (56 c^2 d^2+21 b^2 e^2-2 c e (31 b d+8 a e)\right )-\frac{9}{2} c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{42 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac{e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac{\left ((2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac{e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{256 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac{e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2048 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac{e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac{3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac{e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{2048 \left (c d^2-b d e+a e^2\right )^{11/2}}\\ \end{align*}
Mathematica [A] time = 6.06938, size = 687, normalized size = 1.35 \[ -\frac{(a+x (b+c x))^{3/2} \left (-\frac{-\frac{\left (a+b x+c x^2\right )^{5/2} \left (\frac{3}{4} e \left (-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right )+\frac{9}{2} c d e (2 c d-b e)\right )}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}-\frac{\left (b \left (\frac{3}{4} e \left (-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right )-\frac{9}{2} c d e (2 c d-b e)\right )-2 \left (\frac{3}{4} c d \left (-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right )-\frac{9}{2} a c e^2 (2 c d-b e)\right )\right ) \left (\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac{3 \left (b^2-4 a c\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-x (2 c d-b e)-b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \sqrt{a e^2-b d e+c d^2} \left (4 a e^2-4 b d e+4 c d^2\right )}+\frac{\sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (\frac{1}{2} e (9 b e-14 c d)-2 c d e\right )}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )}\right )}{7 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e \left (a+b x+c x^2\right ) (a+x (b+c x))^{3/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.275, size = 35234, normalized size = 69.1 \begin{align*} \text{output 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 [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]
________________________________________________________________________________________
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.
[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]