Optimal. Leaf size=353 \[ -\frac{2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )-b^2 c (28 a j+b i)+8 c^3 (b g-a h)-4 b^4 j+16 c^4 f\right )+4 b c^2 \left (8 a^2 j-a c h+2 c^2 f\right )+24 a^2 c^3 i+2 b^2 c^2 (3 a i+2 c g)+b^3 c (10 a j+c h)+b^4 c i+b^5 j\right )}{3 c^3 \left (4 a c+b^2\right )^2 \sqrt{a+b x-c x^2}}+\frac{2 \left (x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )+b^2 c (4 a j+b i)+c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j-a c h+c^2 f\right )+a b^2 c i+a b^3 j+2 a c^2 (a i+c g)\right )}{3 c^3 \left (4 a c+b^2\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{j \tan ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c} \sqrt{a+b x-c x^2}}\right )}{c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.3857, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1660, 12, 621, 204} \[ -\frac{2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )-b^2 c (28 a j+b i)+8 c^3 (b g-a h)-4 b^4 j+16 c^4 f\right )+4 b c^2 \left (8 a^2 j-a c h+2 c^2 f\right )+24 a^2 c^3 i+2 b^2 c^2 (3 a i+2 c g)+b^3 c (10 a j+c h)+b^4 c i+b^5 j\right )}{3 c^3 \left (4 a c+b^2\right )^2 \sqrt{a+b x-c x^2}}+\frac{2 \left (x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )+b^2 c (4 a j+b i)+c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j-a c h+c^2 f\right )+a b^2 c i+a b^3 j+2 a c^2 (a i+c g)\right )}{3 c^3 \left (4 a c+b^2\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{j \tan ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c} \sqrt{a+b x-c x^2}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 12
Rule 621
Rule 204
Rubi steps
\begin{align*} \int \frac{f+g x+h x^2+366 x^3+j x^4}{\left (a+b x-c x^2\right )^{5/2}} \, dx &=-\frac{2 \left (c^3 \left (b f-\frac{3 a^2 (244 c+b j)}{c^2}-\frac{a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{2 \int \frac{-\frac{366 b^3 c+4 b c^3 g+b^4 j+b^2 c (c h+a j)+4 c^2 \left (2 c^2 f-a c h-a^2 j\right )}{2 c^3}+\frac{3 \left (b^2+4 a c\right ) (366 c+b j) x}{2 c^2}+\frac{3 \left (b^2+4 a c\right ) j x^2}{2 c}}{\left (a+b x-c x^2\right )^{3/2}} \, dx}{3 \left (b^2+4 a c\right )}\\ &=-\frac{2 \left (c^3 \left (b f-\frac{3 a^2 (244 c+b j)}{c^2}-\frac{a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt{a+b x-c x^2}}+\frac{4 \int \frac{3 \left (b^2+4 a c\right )^2 j}{4 c^2 \sqrt{a+b x-c x^2}} \, dx}{3 \left (b^2+4 a c\right )^2}\\ &=-\frac{2 \left (c^3 \left (b f-\frac{3 a^2 (244 c+b j)}{c^2}-\frac{a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt{a+b x-c x^2}}+\frac{j \int \frac{1}{\sqrt{a+b x-c x^2}} \, dx}{c^2}\\ &=-\frac{2 \left (c^3 \left (b f-\frac{3 a^2 (244 c+b j)}{c^2}-\frac{a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt{a+b x-c x^2}}+\frac{(2 j) \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 )}{c^2}\\ &=-\frac{2 \left (c^3 \left (b f-\frac{3 a^2 (244 c+b j)}{c^2}-\frac{a \left (366 b^2 c+2 c^3 g+b c^2 h+b^3 j\right )}{c^3}\right )-\left (366 b^3 c+b c^2 (1098 a+c g)+b^4 j+b^2 c (c h+4 a j)+2 c^2 \left (c^2 f+a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right ) \left (a+b x-c x^2\right )^{3/2}}-\frac{2 \left (366 b^4 c+8784 a^2 c^3+4 b^2 c^2 (549 a+c g)+b^5 j+b^3 c (c h+10 a j)+4 b c^2 \left (2 c^2 f-a c h+8 a^2 j\right )+2 c \left (183 b^3 c+4 b c^2 (549 a-c g)+2 b^4 j-b^2 c (c h-14 a j)-4 c^2 \left (2 c^2 f-a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2+4 a c\right )^2 \sqrt{a+b x-c x^2}}-\frac{j \tan ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c} \sqrt{a+b x-c x^2}}\right )}{c^{5/2}}\\ \end{align*}
Mathematica [C] time = 1.17812, size = 319, normalized size = 0.9 \[ -\frac{2 \left (b^3 \left (3 a^2 j+18 a c j x^2+c^2 \left (f+3 g x+x^2 (-(3 h+i x))\right )\right )+2 b^2 c \left (21 a^2 j x+a c \left (g+x \left (-6 h+3 i x-14 j x^2\right )\right )+c^2 x (3 f+x (h x-6 g))\right )+4 b c \left (-2 a^2 c (h-3 i x)+5 a^3 j+3 a c^2 \left (f-x \left (g-h x+i x^2\right )\right )+2 c^3 x^2 (g x-3 f)\right )+8 c^2 \left (-a^2 c \left (g+x^2 (3 i+4 j x)\right )+a^3 (2 i+3 j x)-a c^2 x \left (3 f+h x^2\right )+2 c^3 f x^3\right )+b^4 \left (6 a j x-4 c j x^3\right )+3 b^5 j x^2\right )}{3 c^2 \left (4 a c+b^2\right )^2 (a+x (b-c x))^{3/2}}+\frac{i j \log \left (2 \sqrt{a+x (b-c x)}+\frac{i (b-2 c x)}{\sqrt{c}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.059, size = 1453, normalized size = 4.1 \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 [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 [A] time = 1.1967, size = 659, normalized size = 1.87 \begin{align*} -\frac{2 \, \sqrt{-c x^{2} + b x + a}{\left ({\left ({\left (\frac{{\left (16 \, c^{5} f + 8 \, b c^{4} g + 2 \, b^{2} c^{3} h - 8 \, a c^{4} h - b^{3} c^{2} i - 12 \, a b c^{3} i - 4 \, b^{4} c j - 28 \, a b^{2} c^{2} j - 32 \, a^{2} c^{3} j\right )} x}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} - \frac{3 \,{\left (8 \, b c^{4} f + 4 \, b^{2} c^{3} g + b^{3} c^{2} h - 4 \, a b c^{3} h - 2 \, a b^{2} c^{2} i + 8 \, a^{2} c^{3} i - b^{5} j - 6 \, a b^{3} c j\right )}}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c^{3} f - 8 \, a c^{4} f + b^{3} c^{2} g - 4 \, a b c^{3} g - 4 \, a b^{2} c^{2} h + 8 \, a^{2} b c^{2} i + 2 \, a b^{4} j + 14 \, a^{2} b^{2} c j + 8 \, a^{3} c^{2} j\right )}}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{3} c^{2} f + 12 \, a b c^{3} f + 2 \, a b^{2} c^{2} g - 8 \, a^{2} c^{3} g - 8 \, a^{2} b c^{2} h + 16 \, a^{3} c^{2} i + 3 \, a^{2} b^{3} j + 20 \, a^{3} b c j}{b^{4} c^{2} + 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} - b x - a\right )}^{2}} - \frac{j \log \left ({\left | 2 \,{\left (\sqrt{-c} x - \sqrt{-c x^{2} + b x + a}\right )} \sqrt{-c} + b \right |}\right )}{\sqrt{-c} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]