Optimal. Leaf size=849 \[ \frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]
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Rubi [A] time = 2.88348, antiderivative size = 849, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {737, 823, 827, 1169, 634, 618, 206, 628} \[ \frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]
Antiderivative was successfully verified.
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Rule 737
Rule 823
Rule 827
Rule 1169
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\left (a+c x^2\right )^3} \, dx &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}-\frac{\int \frac{-3 d-\frac{5 e x}{2}}{\sqrt{d+e x} \left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\int \frac{\frac{1}{4} c d \left (12 c d^2+13 a e^2\right )+\frac{1}{4} c e \left (6 c d^2+5 a e^2\right ) x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )+\frac{1}{4} c e \left (6 c d^2+5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{4 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac{1}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac{1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac{1}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac{1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}+\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt{d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}-\frac{\left (e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (d-\frac{\sqrt{c d^2+a e^2}}{\sqrt{c}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt{d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{x \sqrt{d+e x}}{4 a \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt{2} \sqrt{d+e x}\right )}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt{2} \sqrt{d+e x}\right )}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c d^2+a e^2}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}
Mathematica [A] time = 0.870625, size = 412, normalized size = 0.49 \[ \frac{\frac{2 (d+e x)^{3/2} \left (5 a^2 e^3+a c d e (3 d+8 e x)+6 c^2 d^3 x\right )}{a+c x^2}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (5 a^2 e^4+6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 \sqrt{-a} a \sqrt{c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (5 a^2 e^4-6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 (-a)^{3/2} \sqrt{c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )-4 \sqrt{-a} c^{3/4} d e \sqrt{d+e x} \left (4 a e^2+3 c d^2\right )}{\sqrt{-a} c^{3/4}}+\frac{8 a (d+e x)^{3/2} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{32 a^2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}}\sqrt{ex+d}}\, dx \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{\sqrt{e x + d}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.63693, size = 7772, normalized size = 9.15 \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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