Optimal. Leaf size=172 \[ \frac {e^{3/2} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{5/2} d^{5/2}}-\frac {2 e (d+e x)}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {668, 652, 621, 206} \begin {gather*} -\frac {2 e (d+e x)}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {e^{3/2} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{5/2} d^{5/2}}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 652
Rule 668
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {e \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d}\\ &=-\frac {2 (d+e x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 e (d+e x)}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {e^2 \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c^2 d^2}\\ &=-\frac {2 (d+e x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 e (d+e x)}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^2 d^2}\\ &=-\frac {2 (d+e x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 e (d+e x)}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {e^{3/2} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 98, normalized size = 0.57 \begin {gather*} -\frac {2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{3 c d (a e+c d x)^3 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.01, size = 469, normalized size = 2.73 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{6 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{3 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 620, normalized size = 3.60 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {4 \, {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}} + \frac {3 \, {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )} x + \frac {6 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )} x + \frac {c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )}}{3 \, {\left (c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}} - \frac {\sqrt {c d} e^{\frac {3}{2}} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 2538, normalized size = 14.76
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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