Optimal. Leaf size=216 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}+\frac {B e^4 \log \left (a+c x^2\right )}{2 c^3} \]
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Rubi [A] time = 0.21, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {819, 635, 205, 260} \begin {gather*} -\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}+\frac {B e^4 \log \left (a+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 819
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx &=-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {(d+e x)^2 \left (3 A c d^2+a e (4 B d+3 A e)+4 a B e^2 x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e \left (8 a B d e+3 A \left (c d^2+a e^2\right )\right )+\left (4 a^2 B e^3-c d \left (3 A c d^2+a e (4 B d+3 A e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {3 A \left (c d^2+a e^2\right )^2+4 a B d e \left (c d^2+3 a e^2\right )+8 a^2 B e^4 x}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e \left (8 a B d e+3 A \left (c d^2+a e^2\right )\right )+\left (4 a^2 B e^3-c d \left (3 A c d^2+a e (4 B d+3 A e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (B e^4\right ) \int \frac {x}{a+c x^2} \, dx}{c^2}+\frac {\left (3 A \left (c d^2+a e^2\right )^2+4 a B d e \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e \left (8 a B d e+3 A \left (c d^2+a e^2\right )\right )+\left (4 a^2 B e^3-c d \left (3 A c d^2+a e (4 B d+3 A e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 A \left (c d^2+a e^2\right )^2+4 a B d e \left (c d^2+3 a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {B e^4 \log \left (a+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 263, normalized size = 1.22 \begin {gather*} \frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{a^{5/2}}+\frac {-2 a^3 B e^4+2 a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-2 a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))+2 A c^3 d^4 x}{a \left (a+c x^2\right )^2}+\frac {8 a^3 B e^4-a^2 c e^2 (A e (16 d+5 e x)+4 B d (6 d+5 e x))+2 a c^2 d^2 e x (3 A e+2 B d)+3 A c^3 d^4 x}{a^2 \left (a+c x^2\right )}+4 B e^4 \log \left (a+c x^2\right )}{8 c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.45, size = 1055, normalized size = 4.88 \begin {gather*} \left [-\frac {4 \, B a^{3} c^{2} d^{4} + 16 \, A a^{3} c^{2} d^{3} e + 24 \, B a^{4} c d^{2} e^{2} + 16 \, A a^{4} c d e^{3} - 12 \, B a^{5} e^{4} - 2 \, {\left (3 \, A a c^{4} d^{4} + 4 \, B a^{2} c^{3} d^{3} e + 6 \, A a^{2} c^{3} d^{2} e^{2} - 20 \, B a^{3} c^{2} d e^{3} - 5 \, A a^{3} c^{2} e^{4}\right )} x^{3} + 16 \, {\left (3 \, B a^{3} c^{2} d^{2} e^{2} + 2 \, A a^{3} c^{2} d e^{3} - B a^{4} c e^{4}\right )} x^{2} + {\left (3 \, A a^{2} c^{2} d^{4} + 4 \, B a^{3} c d^{3} e + 6 \, A a^{3} c d^{2} e^{2} + 12 \, B a^{4} d e^{3} + 3 \, A a^{4} e^{4} + {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} + 12 \, B a^{2} c^{2} d e^{3} + 3 \, A a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (3 \, A a c^{3} d^{4} + 4 \, B a^{2} c^{2} d^{3} e + 6 \, A a^{2} c^{2} d^{2} e^{2} + 12 \, B a^{3} c d e^{3} + 3 \, A a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (5 \, A a^{2} c^{3} d^{4} - 4 \, B a^{3} c^{2} d^{3} e - 6 \, A a^{3} c^{2} d^{2} e^{2} - 12 \, B a^{4} c d e^{3} - 3 \, A a^{4} c e^{4}\right )} x - 8 \, {\left (B a^{3} c^{2} e^{4} x^{4} + 2 \, B a^{4} c e^{4} x^{2} + B a^{5} e^{4}\right )} \log \left (c x^{2} + a\right )}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{4} + 8 \, A a^{3} c^{2} d^{3} e + 12 \, B a^{4} c d^{2} e^{2} + 8 \, A a^{4} c d e^{3} - 6 \, B a^{5} e^{4} - {\left (3 \, A a c^{4} d^{4} + 4 \, B a^{2} c^{3} d^{3} e + 6 \, A a^{2} c^{3} d^{2} e^{2} - 20 \, B a^{3} c^{2} d e^{3} - 5 \, A a^{3} c^{2} e^{4}\right )} x^{3} + 8 \, {\left (3 \, B a^{3} c^{2} d^{2} e^{2} + 2 \, A a^{3} c^{2} d e^{3} - B a^{4} c e^{4}\right )} x^{2} - {\left (3 \, A a^{2} c^{2} d^{4} + 4 \, B a^{3} c d^{3} e + 6 \, A a^{3} c d^{2} e^{2} + 12 \, B a^{4} d e^{3} + 3 \, A a^{4} e^{4} + {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} + 12 \, B a^{2} c^{2} d e^{3} + 3 \, A a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (3 \, A a c^{3} d^{4} + 4 \, B a^{2} c^{2} d^{3} e + 6 \, A a^{2} c^{2} d^{2} e^{2} + 12 \, B a^{3} c d e^{3} + 3 \, A a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{3} d^{4} - 4 \, B a^{3} c^{2} d^{3} e - 6 \, A a^{3} c^{2} d^{2} e^{2} - 12 \, B a^{4} c d e^{3} - 3 \, A a^{4} c e^{4}\right )} x - 4 \, {\left (B a^{3} c^{2} e^{4} x^{4} + 2 \, B a^{4} c e^{4} x^{2} + B a^{5} e^{4}\right )} \log \left (c x^{2} + a\right )}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 312, normalized size = 1.44 \begin {gather*} \frac {B e^{4} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} + 12 \, B a^{2} d e^{3} + 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (3 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 20 \, B a^{2} c d e^{3} - 5 \, A a^{2} c e^{4}\right )} x^{3} - 8 \, {\left (3 \, B a^{2} c d^{2} e^{2} + 2 \, A a^{2} c d e^{3} - B a^{3} e^{4}\right )} x^{2} + {\left (5 \, A a c^{2} d^{4} - 4 \, B a^{2} c d^{3} e - 6 \, A a^{2} c d^{2} e^{2} - 12 \, B a^{3} d e^{3} - 3 \, A a^{3} e^{4}\right )} x - \frac {2 \, {\left (B a^{2} c^{2} d^{4} + 4 \, A a^{2} c^{2} d^{3} e + 6 \, B a^{3} c d^{2} e^{2} + 4 \, A a^{3} c d e^{3} - 3 \, B a^{4} e^{4}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 359, normalized size = 1.66 \begin {gather*} \frac {3 A \,d^{2} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, a c}+\frac {3 A \,d^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {3 A \,e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, c^{2}}+\frac {B \,d^{3} e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a c}+\frac {3 B d \,e^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}+\frac {B \,e^{4} \ln \left (c \,x^{2}+a \right )}{2 c^{3}}+\frac {-\frac {\left (2 A c d e -B a \,e^{2}+3 B c \,d^{2}\right ) e^{2} x^{2}}{c^{2}}-\frac {\left (5 A \,a^{2} e^{4}-6 A a c \,d^{2} e^{2}-3 A \,c^{2} d^{4}+20 B \,a^{2} d \,e^{3}-4 B a c \,d^{3} e \right ) x^{3}}{8 a^{2} c}-\frac {\left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}-5 A \,c^{2} d^{4}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right ) x}{8 a \,c^{2}}-\frac {4 A d a c \,e^{3}+4 A \,c^{2} d^{3} e -3 B \,a^{2} e^{4}+6 a \,d^{2} B \,e^{2} c +B \,c^{2} d^{4}}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 352, normalized size = 1.63 \begin {gather*} \frac {B e^{4} \log \left (c x^{2} + a\right )}{2 \, c^{3}} - \frac {2 \, B a^{2} c^{2} d^{4} + 8 \, A a^{2} c^{2} d^{3} e + 12 \, B a^{3} c d^{2} e^{2} + 8 \, A a^{3} c d e^{3} - 6 \, B a^{4} e^{4} - {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} - 20 \, B a^{2} c^{2} d e^{3} - 5 \, A a^{2} c^{2} e^{4}\right )} x^{3} + 8 \, {\left (3 \, B a^{2} c^{2} d^{2} e^{2} + 2 \, A a^{2} c^{2} d e^{3} - B a^{3} c e^{4}\right )} x^{2} - {\left (5 \, A a c^{3} d^{4} - 4 \, B a^{2} c^{2} d^{3} e - 6 \, A a^{2} c^{2} d^{2} e^{2} - 12 \, B a^{3} c d e^{3} - 3 \, A a^{3} c e^{4}\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {{\left (3 \, A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} + 12 \, B a^{2} d e^{3} + 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.56, size = 763, normalized size = 3.53 \begin {gather*} \frac {5\,A\,d^4\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^4}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,B\,a^2\,e^4}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {A\,d^3\,e}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}-\frac {5\,A\,e^4\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {B\,e^4\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {A\,a\,d\,e^3}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}+\frac {3\,A\,c\,d^4\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}-\frac {3\,A\,a\,e^4\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {3\,A\,d^2\,e^2\,x^3}{4\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {3\,A\,d^2\,e^2\,x}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {2\,A\,d\,e^3\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}-\frac {5\,B\,d\,e^3\,x^3}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,B\,a\,d^2\,e^2}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {B\,a\,e^4\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}-\frac {3\,B\,d^2\,e^2\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}+\frac {3\,A\,d^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,A\,e^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}+\frac {B\,d^3\,e\,x^3}{2\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^3\,e\,x}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,B\,d\,e^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{5/2}}+\frac {B\,d^3\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,c^{3/2}}+\frac {3\,A\,d^2\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{4\,a^{3/2}\,c^{3/2}}-\frac {3\,B\,a\,d\,e^3\,x}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 46.14, size = 816, normalized size = 3.78 \begin {gather*} \left (\frac {B e^{4}}{2 c^{3}} - \frac {\sqrt {- a^{5} c^{7}} \left (3 A a^{2} e^{4} + 6 A a c d^{2} e^{2} + 3 A c^{2} d^{4} + 12 B a^{2} d e^{3} + 4 B a c d^{3} e\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {- 8 B a^{3} e^{4} + 16 a^{3} c^{3} \left (\frac {B e^{4}}{2 c^{3}} - \frac {\sqrt {- a^{5} c^{7}} \left (3 A a^{2} e^{4} + 6 A a c d^{2} e^{2} + 3 A c^{2} d^{4} + 12 B a^{2} d e^{3} + 4 B a c d^{3} e\right )}{16 a^{5} c^{6}}\right )}{3 A a^{2} c e^{4} + 6 A a c^{2} d^{2} e^{2} + 3 A c^{3} d^{4} + 12 B a^{2} c d e^{3} + 4 B a c^{2} d^{3} e} \right )} + \left (\frac {B e^{4}}{2 c^{3}} + \frac {\sqrt {- a^{5} c^{7}} \left (3 A a^{2} e^{4} + 6 A a c d^{2} e^{2} + 3 A c^{2} d^{4} + 12 B a^{2} d e^{3} + 4 B a c d^{3} e\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {- 8 B a^{3} e^{4} + 16 a^{3} c^{3} \left (\frac {B e^{4}}{2 c^{3}} + \frac {\sqrt {- a^{5} c^{7}} \left (3 A a^{2} e^{4} + 6 A a c d^{2} e^{2} + 3 A c^{2} d^{4} + 12 B a^{2} d e^{3} + 4 B a c d^{3} e\right )}{16 a^{5} c^{6}}\right )}{3 A a^{2} c e^{4} + 6 A a c^{2} d^{2} e^{2} + 3 A c^{3} d^{4} + 12 B a^{2} c d e^{3} + 4 B a c^{2} d^{3} e} \right )} + \frac {- 8 A a^{3} c d e^{3} - 8 A a^{2} c^{2} d^{3} e + 6 B a^{4} e^{4} - 12 B a^{3} c d^{2} e^{2} - 2 B a^{2} c^{2} d^{4} + x^{3} \left (- 5 A a^{2} c^{2} e^{4} + 6 A a c^{3} d^{2} e^{2} + 3 A c^{4} d^{4} - 20 B a^{2} c^{2} d e^{3} + 4 B a c^{3} d^{3} e\right ) + x^{2} \left (- 16 A a^{2} c^{2} d e^{3} + 8 B a^{3} c e^{4} - 24 B a^{2} c^{2} d^{2} e^{2}\right ) + x \left (- 3 A a^{3} c e^{4} - 6 A a^{2} c^{2} d^{2} e^{2} + 5 A a c^{3} d^{4} - 12 B a^{3} c d e^{3} - 4 B a^{2} c^{2} d^{3} e\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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