Optimal. Leaf size=366 \[ \frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]
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Rubi [A] time = 0.279515, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {662, 672, 660, 205} \[ \frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 672
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{(c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx}{2 e}\\ &=-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{\left (3 c^2 d^2\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx}{16 e^2}\\ &=-\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{\left (c^3 d^3\right ) \int \frac{1}{(d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 e^3}\\ &=-\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{\left (3 c^4 d^4\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 e^3 \left (c d^2-a e^2\right )}\\ &=-\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^4 d^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{\left (3 c^5 d^5\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 e^3 \left (c d^2-a e^2\right )^2}\\ &=-\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^4 d^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{\left (3 c^5 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{128 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^4 d^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d^2-a e^2} \sqrt{d+e x}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0747819, size = 83, normalized size = 0.23 \[ \frac{2 c^5 d^5 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.278, size = 910, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{17}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30276, size = 3484, normalized size = 9.52 \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 [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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