Optimal. Leaf size=161 \[ -\frac {d e \sqrt {a+c x^2} \left (5 a e^2+2 c d^2\right )}{3 a^2 c^2}-\frac {(d+e x) \left (a e \left (3 a e^2+c d^2\right )-2 c d x \left (2 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {(d+e x)^3 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {739, 819, 641, 217, 206} \[ -\frac {d e \sqrt {a+c x^2} \left (5 a e^2+2 c d^2\right )}{3 a^2 c^2}-\frac {(d+e x) \left (a e \left (3 a e^2+c d^2\right )-2 c d x \left (2 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {(d+e x)^3 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rule 739
Rule 819
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(d+e x)^2 \left (2 c d^2+3 a e^2-c d e x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {3 a^2 e^4-c d e \left (2 c d^2+5 a e^2\right ) x}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2}\\ &=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c^2}\\ &=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 130, normalized size = 0.81 \[ \frac {-a^3 e^3 (8 d+3 e x)-4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )+3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )+2 c^3 d^4 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}}+\frac {e^4 \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 401, normalized size = 2.49 \[ \left [\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (12 \, a^{2} c^{2} d e^{3} x^{2} + 4 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} - 2 \, {\left (c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} - 2 \, a^{2} c^{2} e^{4}\right )} x^{3} - 3 \, {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}, -\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (12 \, a^{2} c^{2} d e^{3} x^{2} + 4 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} - 2 \, {\left (c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} - 2 \, a^{2} c^{2} e^{4}\right )} x^{3} - 3 \, {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 150, normalized size = 0.93 \[ -\frac {{\left (2 \, x {\left (\frac {6 \, d e^{3}}{c} - \frac {{\left (c^{5} d^{4} + 3 \, a c^{4} d^{2} e^{2} - 2 \, a^{2} c^{3} e^{4}\right )} x}{a^{2} c^{4}}\right )} - \frac {3 \, {\left (a c^{4} d^{4} - a^{3} c^{2} e^{4}\right )}}{a^{2} c^{4}}\right )} x + \frac {4 \, {\left (a^{2} c^{3} d^{3} e + 2 \, a^{3} c^{2} d e^{3}\right )}}{a^{2} c^{4}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {e^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 202, normalized size = 1.25 \[ -\frac {e^{4} x^{3}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {4 d \,e^{3} x^{2}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} c}+\frac {d^{4} x}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a}-\frac {2 d^{2} e^{2} x}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {8 a d \,e^{3}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{2}}+\frac {2 d^{2} e^{2} x}{\sqrt {c \,x^{2}+a}\, a c}+\frac {2 d^{4} x}{3 \sqrt {c \,x^{2}+a}\, a^{2}}-\frac {4 d^{3} e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {e^{4} x}{\sqrt {c \,x^{2}+a}\, c^{2}}+\frac {e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 213, normalized size = 1.32 \[ -\frac {1}{3} \, e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} - \frac {4 \, d e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{4} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{4} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {2 \, d^{2} e^{2} x}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{2} e^{2} x}{\sqrt {c x^{2} + a} a c} - \frac {e^{4} x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {4 \, d^{3} e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {8 \, a d e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{4}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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