Optimal. Leaf size=234 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^{7/2} c^{7/2}}-\frac{(d+e x) (a e-c d x) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^3 c^3 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.291445, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1645, 805, 723, 205} \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^{7/2} c^{7/2}}-\frac{(d+e x) (a e-c d x) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^3 c^3 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 805
Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{\int \frac{(d+e x)^3 (-5 A c d-a C d-4 a B e-(A c+5 a C) e x)}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) (a e-c d x) (d+e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{\left (\left (c d^2+a e^2\right ) \left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 c^3}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) (a e-c d x) (d+e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{\left (c d^2+a e^2\right ) \left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.305285, size = 437, normalized size = 1.87 \[ \frac{a^2 c e^2 x \left (e (A e+4 B d)+6 C d^2\right )-a^3 e^3 (8 B e+32 C d+11 C e x)+a c^2 d^2 x \left (6 A e^2+4 B d e+C d^2\right )+5 A c^3 d^4 x}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{a^2 c e \left (e (A e (4 d+e x)+2 B d (3 d+2 e x))+2 C d^2 (2 d+3 e x)\right )-a^3 e^3 (B e+4 C d+C e x)-a c^2 d^2 \left (4 A d e+6 A e^2 x+B d (d+4 e x)+C d^2 x\right )+A c^3 d^4 x}{6 a c^3 \left (a+c x^2\right )^3}+\frac{-a^2 c e \left (e (A e (24 d+7 e x)+4 B d (9 d+7 e x))+6 C d^2 (4 d+7 e x)\right )+a^3 e^3 (12 B e+48 C d+13 C e x)+a c^2 d^2 x \left (6 A e^2+4 B d e+C d^2\right )+5 A c^3 d^4 x}{24 a^2 c^3 \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (a e^2+5 c d^2\right )+a \left (5 a C e^2+c d (4 B e+C d)\right )\right )}{16 a^{7/2} c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 647, normalized size = 2.8 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ( A{e}^{4}{a}^{2}c+6\,Aa{c}^{2}{d}^{2}{e}^{2}+5\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c+4\,Ba{c}^{2}{d}^{3}e-11\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}+Ca{c}^{2}{d}^{4} \right ){x}^{5}}{16\,{a}^{3}c}}-{\frac{{e}^{3} \left ( Be+4\,Cd \right ){x}^{4}}{2\,c}}-{\frac{ \left ( A{e}^{4}{a}^{2}c-6\,Aa{c}^{2}{d}^{2}{e}^{2}-5\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c-4\,Ba{c}^{2}{d}^{3}e+5\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}-Ca{c}^{2}{d}^{4} \right ){x}^{3}}{6\,{a}^{2}{c}^{2}}}-{\frac{e \left ( 2\,Acd{e}^{2}+Ba{e}^{3}+3\,Bc{d}^{2}e+4\,Cad{e}^{2}+2\,Cc{d}^{3} \right ){x}^{2}}{2\,{c}^{2}}}-{\frac{ \left ( A{e}^{4}{a}^{2}c+6\,Aa{c}^{2}{d}^{2}{e}^{2}-11\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c+4\,Ba{c}^{2}{d}^{3}e+5\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}+Ca{c}^{2}{d}^{4} \right ) x}{16\,a{c}^{3}}}-{\frac{2\,Aacd{e}^{3}+4\,A{c}^{2}{d}^{3}e+B{a}^{2}{e}^{4}+3\,Bac{d}^{2}{e}^{2}+B{c}^{2}{d}^{4}+4\,C{a}^{2}d{e}^{3}+2\,Cac{d}^{3}e}{6\,{c}^{3}}} \right ) }+{\frac{A{e}^{4}}{16\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{2}{e}^{2}}{8\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,A{d}^{4}}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bd{e}^{3}}{4\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{d}^{3}e}{4\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,C{e}^{4}}{16\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,C{d}^{2}{e}^{2}}{8\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{4}}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 [B] time = 1.71336, size = 3753, normalized size = 16.04 \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 [B] time = 1.16659, size = 859, normalized size = 3.67 \begin{align*} \frac{{\left (C a c^{2} d^{4} + 5 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, C a^{2} c d^{2} e^{2} + 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + 5 \, C a^{3} e^{4} + A a^{2} c e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c^{3}} + \frac{3 \, C a c^{4} d^{4} x^{5} + 15 \, A c^{5} d^{4} x^{5} + 12 \, B a c^{4} d^{3} x^{5} e + 18 \, C a^{2} c^{3} d^{2} x^{5} e^{2} + 18 \, A a c^{4} d^{2} x^{5} e^{2} + 8 \, C a^{2} c^{3} d^{4} x^{3} + 40 \, A a c^{4} d^{4} x^{3} + 12 \, B a^{2} c^{3} d x^{5} e^{3} + 32 \, B a^{2} c^{3} d^{3} x^{3} e - 33 \, C a^{3} c^{2} x^{5} e^{4} + 3 \, A a^{2} c^{3} x^{5} e^{4} - 96 \, C a^{3} c^{2} d x^{4} e^{3} - 48 \, C a^{3} c^{2} d^{2} x^{3} e^{2} + 48 \, A a^{2} c^{3} d^{2} x^{3} e^{2} - 48 \, C a^{3} c^{2} d^{3} x^{2} e - 3 \, C a^{3} c^{2} d^{4} x + 33 \, A a^{2} c^{3} d^{4} x - 24 \, B a^{3} c^{2} x^{4} e^{4} - 32 \, B a^{3} c^{2} d x^{3} e^{3} - 72 \, B a^{3} c^{2} d^{2} x^{2} e^{2} - 12 \, B a^{3} c^{2} d^{3} x e - 8 \, B a^{3} c^{2} d^{4} - 40 \, C a^{4} c x^{3} e^{4} - 8 \, A a^{3} c^{2} x^{3} e^{4} - 96 \, C a^{4} c d x^{2} e^{3} - 48 \, A a^{3} c^{2} d x^{2} e^{3} - 18 \, C a^{4} c d^{2} x e^{2} - 18 \, A a^{3} c^{2} d^{2} x e^{2} - 16 \, C a^{4} c d^{3} e - 32 \, A a^{3} c^{2} d^{3} e - 24 \, B a^{4} c x^{2} e^{4} - 12 \, B a^{4} c d x e^{3} - 24 \, B a^{4} c d^{2} e^{2} - 15 \, C a^{5} x e^{4} - 3 \, A a^{4} c x e^{4} - 32 \, C a^{5} d e^{3} - 16 \, A a^{4} c d e^{3} - 8 \, B a^{5} e^{4}}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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