Optimal. Leaf size=91 \[ \frac {16 d x}{35 a^4 \sqrt {a+c x^2}}+\frac {8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {c d x-a e}{7 a c \left (a+c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {639, 192, 191} \begin {gather*} \frac {16 d x}{35 a^4 \sqrt {a+c x^2}}+\frac {8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}-\frac {a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 639
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^2\right )^{9/2}} \, dx &=-\frac {a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac {(6 d) \int \frac {1}{\left (a+c x^2\right )^{7/2}} \, dx}{7 a}\\ &=-\frac {a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {(24 d) \int \frac {1}{\left (a+c x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=-\frac {a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {(16 d) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=-\frac {a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 d x}{35 a^4 \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.74 \begin {gather*} \frac {-5 a^4 e+35 a^3 c d x+70 a^2 c^2 d x^3+56 a c^3 d x^5+16 c^4 d x^7}{35 a^4 c \left (a+c x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 67, normalized size = 0.74 \begin {gather*} \frac {-5 a^4 e+35 a^3 c d x+70 a^2 c^2 d x^3+56 a c^3 d x^5+16 c^4 d x^7}{35 a^4 c \left (a+c x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 108, normalized size = 1.19 \begin {gather*} \frac {{\left (16 \, c^{4} d x^{7} + 56 \, a c^{3} d x^{5} + 70 \, a^{2} c^{2} d x^{3} + 35 \, a^{3} c d x - 5 \, a^{4} e\right )} \sqrt {c x^{2} + a}}{35 \, {\left (a^{4} c^{5} x^{8} + 4 \, a^{5} c^{4} x^{6} + 6 \, a^{6} c^{3} x^{4} + 4 \, a^{7} c^{2} x^{2} + a^{8} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 68, normalized size = 0.75 \begin {gather*} \frac {{\left (2 \, {\left (4 \, {\left (\frac {2 \, c^{3} d x^{2}}{a^{4}} + \frac {7 \, c^{2} d}{a^{3}}\right )} x^{2} + \frac {35 \, c d}{a^{2}}\right )} x^{2} + \frac {35 \, d}{a}\right )} x - \frac {5 \, e}{c}}{35 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 64, normalized size = 0.70 \begin {gather*} -\frac {-16 c^{4} d \,x^{7}-56 c^{3} d \,x^{5} a -70 c^{2} d \,x^{3} a^{2}-35 d x \,a^{3} c +5 e \,a^{4}}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 80, normalized size = 0.88 \begin {gather*} \frac {16 \, d x}{35 \, \sqrt {c x^{2} + a} a^{4}} + \frac {8 \, d x}{35 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, d x}{35 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {d x}{7 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {e}{7 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 74, normalized size = 0.81 \begin {gather*} \frac {16\,d\,x}{35\,a^4\,\sqrt {c\,x^2+a}}-\frac {\frac {e}{7\,c}-\frac {d\,x}{7\,a}}{{\left (c\,x^2+a\right )}^{7/2}}+\frac {8\,d\,x}{35\,a^3\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {6\,d\,x}{35\,a^2\,{\left (c\,x^2+a\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 39.86, size = 1360, normalized size = 14.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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