Optimal. Leaf size=226 \[ -\frac {\sqrt {c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}-\frac {\left (a e^2+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}+\frac {\sqrt {a+c x^2} \left (8 \left (a e^2+c d^2\right )^2-c d e x \left (7 a e^2+4 c d^2\right )\right )}{8 e^5}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a e^2+c d^2\right )-3 c d e x\right )}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e} \]
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Rubi [A] time = 0.26, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {735, 815, 844, 217, 206, 725} \begin {gather*} -\frac {\sqrt {c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a e^2+c d^2\right )-3 c d e x\right )}{12 e^3}+\frac {\sqrt {a+c x^2} \left (8 \left (a e^2+c d^2\right )^2-c d e x \left (7 a e^2+4 c d^2\right )\right )}{8 e^5}-\frac {\left (a e^2+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}+\frac {\left (a+c x^2\right )^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 735
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{d+e x} \, dx &=\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {\left (a c e \left (c d^2+4 a e^2\right )-c^2 d \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{4 c e^3}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {a c^2 e \left (4 c^2 d^4+9 a c d^2 e^2+8 a^2 e^4\right )-c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 c^2 e^5}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}+\frac {\left (c d^2+a e^2\right )^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^6}-\frac {\left (c d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 e^6}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2+a e^2\right )^3 \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^6}-\frac {\left (c d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 e^6}\\ &=\frac {\left (8 \left (c d^2+a e^2\right )^2-c d e \left (4 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5}+\frac {\left (4 \left (c d^2+a e^2\right )-3 c d e x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3}+\frac {\left (a+c x^2\right )^{5/2}}{5 e}-\frac {\sqrt {c} d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 e^6}-\frac {\left (c d^2+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 332, normalized size = 1.47 \begin {gather*} \frac {-\frac {5 \sqrt {c} d \sqrt {a+c x^2} \left (3 a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+\sqrt {c} x \left (5 a+2 c x^2\right ) \sqrt {\frac {c x^2}{a}+1}\right )}{8 e \sqrt {\frac {c x^2}{a}+1}}-\frac {5 \left (a e^2+c d^2\right ) \left (\sqrt {\frac {c x^2}{a}+1} \left (-e \sqrt {a+c x^2} \left (8 a e^2+6 c d^2-3 c d e x+2 c e^2 x^2\right )+6 \left (a e^2+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+6 \sqrt {c} d \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )\right )+3 \sqrt {a} \sqrt {c} d e^2 \sqrt {a+c x^2} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )}{6 e^5 \sqrt {\frac {c x^2}{a}+1}}+\left (a+c x^2\right )^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.81, size = 275, normalized size = 1.22 \begin {gather*} \frac {\left (15 a^2 \sqrt {c} d e^4+20 a c^{3/2} d^3 e^2+8 c^{5/2} d^5\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 e^6}+\frac {2 \sqrt {-a e^2-c d^2} \left (a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac {-e \sqrt {a+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {-a e^2-c d^2}}\right )}{e^6}+\frac {\sqrt {a+c x^2} \left (184 a^2 e^4+280 a c d^2 e^2-135 a c d e^3 x+88 a c e^4 x^2+120 c^2 d^4-60 c^2 d^3 e x+40 c^2 d^2 e^2 x^2-30 c^2 d e^3 x^3+24 c^2 e^4 x^4\right )}{120 e^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 51.03, size = 1176, normalized size = 5.20
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 282, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, {\left (8 \, c^{\frac {5}{2}} d^{5} + 20 \, a c^{\frac {3}{2}} d^{3} e^{2} + 15 \, a^{2} \sqrt {c} d e^{4}\right )} e^{\left (-6\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (4 \, c^{2} x e^{\left (-1\right )} - 5 \, c^{2} d e^{\left (-2\right )}\right )} x + \frac {4 \, {\left (5 \, c^{5} d^{2} e^{18} + 11 \, a c^{4} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x - \frac {15 \, {\left (4 \, c^{5} d^{3} e^{17} + 9 \, a c^{4} d e^{19}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x + \frac {8 \, {\left (15 \, c^{5} d^{4} e^{16} + 35 \, a c^{4} d^{2} e^{18} + 23 \, a^{2} c^{3} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1225, normalized size = 5.42
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 272, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {c x^{2} + a} c^{2} d^{3} x}{2 \, e^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d x}{4 \, e^{2}} - \frac {7 \, \sqrt {c x^{2} + a} a c d x}{8 \, e^{2}} - \frac {c^{\frac {5}{2}} d^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{6}} - \frac {5 \, a c^{\frac {3}{2}} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, e^{4}} - \frac {15 \, a^{2} \sqrt {c} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, e^{2}} + \frac {{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{e} + \frac {\sqrt {c x^{2} + a} c^{2} d^{4}}{e^{5}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2}}{3 \, e^{3}} + \frac {2 \, \sqrt {c x^{2} + a} a c d^{2}}{e^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{5 \, e} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a}{3 \, e} + \frac {\sqrt {c x^{2} + a} a^{2}}{e} \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{d+e\,x} \,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 (a + c x^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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