Optimal. Leaf size=448 \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]
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Rubi [A] time = 0.888996, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {864, 870, 891, 63, 217, 206} \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 864
Rule 870
Rule 891
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{(c d f-a e g) \int \frac{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{2 g}\\ &=-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}+\frac{\left (3 (c d f-a e g)^2\right ) \int \frac{(f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{16 g^2}\\ &=\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{(c d f-a e g)^3 \int \frac{\sqrt{d+e x} (f+g x)^{3/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 g^3}\\ &=-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^4\right ) \int \frac{\sqrt{d+e x} \sqrt{f+g x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d g^3}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5\right ) \int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 g^3}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \int \frac{1}{\sqrt{a e+c d x} \sqrt{f+g x}} \, dx}{256 c^2 d^2 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{a e g}{c d}+\frac{g x^2}{c d}}} \, dx,x,\sqrt{a e+c d x}\right )}{128 c^3 d^3 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{c d}} \, dx,x,\frac{\sqrt{a e+c d x}}{\sqrt{f+g x}}\right )}{128 c^3 d^3 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [B] time = 6.129, size = 974, normalized size = 2.17 \[ \frac{2 (c d f-a e g) (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \sqrt{f+g x} \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^{5/2} \left (\frac{21 (c d f-a e g)^4 \left (\frac{16 c^3 d^3 g^3 (a e+c d x)^3}{15 (c d f-a e g)^3 \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^3}-\frac{4 c^2 d^2 g^2 (a e+c d x)^2}{3 (c d f-a e g)^2 \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^2}+\frac{2 c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{g} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g} \sqrt{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}}}\right ) \sqrt{a e+c d x}}{\sqrt{c d f-a e g} \sqrt{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}} \sqrt{\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1}}\right ) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^4}{512 c^4 d^4 g^4 (a e+c d x)^4 \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^2}+\frac{7}{10} \left (\frac{1}{\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1}+\frac{3}{8 \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^2}\right )\right )}{7 c^2 d^2 \left (\frac{c d}{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}}\right )^{3/2} (d+e x)^{5/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.41, size = 1191, normalized size = 2.7 \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 (g x + f\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 28.5776, size = 2781, normalized size = 6.21 \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] 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 (g x + f\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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