Optimal. Leaf size=154 \[ \frac {3 a^2 e^3+c d x \left (5 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {741, 823, 12, 725, 206} \begin {gather*} \frac {3 a^2 e^3+c d x \left (5 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 741
Rule 823
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+c x^2\right )^{5/2}} \, dx &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 c d^2-3 a e^2-2 c d e x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {3 a^2 e^3+c d \left (2 c d^2+5 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {\int \frac {3 a^2 c e^4}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {3 a^2 e^3+c d \left (2 c d^2+5 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {e^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {3 a^2 e^3+c d \left (2 c d^2+5 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}-\frac {e^4 \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 )}{\left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {3 a^2 e^3+c d \left (2 c d^2+5 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 153, normalized size = 0.99 \begin {gather*} \frac {4 a^3 e^3+a^2 c e \left (d^2+6 d e x+3 e^2 x^2\right )+a c^2 d x \left (3 d^2+5 e^2 x^2\right )+2 c^3 d^3 x^3}{3 a^2 \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )^2}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.91, size = 224, normalized size = 1.45 \begin {gather*} \frac {4 a^3 e^3+a^2 c d^2 e+6 a^2 c d e^2 x+3 a^2 c e^3 x^2+3 a c^2 d^3 x+5 a c^2 d e^2 x^3+2 c^3 d^3 x^3}{3 a^2 \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )^2}+\frac {2 e^4 \sqrt {-a e^2-c d^2} \tan ^{-1}\left (-\frac {e \sqrt {a+c x^2}}{\sqrt {-a e^2-c d^2}}+\frac {\sqrt {c} e x}{\sqrt {-a e^2-c d^2}}+\frac {\sqrt {c} d}{\sqrt {-a e^2-c d^2}}\right )}{\left (a e^2+c d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 858, normalized size = 5.57 \begin {gather*} \left [\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (a^{2} c^{2} d^{4} e + 5 \, a^{3} c d^{2} e^{3} + 4 \, a^{4} e^{5} + {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} + 3 \, a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{4} c^{3} d^{6} + 3 \, a^{5} c^{2} d^{4} e^{2} + 3 \, a^{6} c d^{2} e^{4} + a^{7} e^{6} + {\left (a^{2} c^{5} d^{6} + 3 \, a^{3} c^{4} d^{4} e^{2} + 3 \, a^{4} c^{3} d^{2} e^{4} + a^{5} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} + 3 \, a^{4} c^{3} d^{4} e^{2} + 3 \, a^{5} c^{2} d^{2} e^{4} + a^{6} c e^{6}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (a^{2} c^{2} d^{4} e + 5 \, a^{3} c d^{2} e^{3} + 4 \, a^{4} e^{5} + {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} + 3 \, a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{4} c^{3} d^{6} + 3 \, a^{5} c^{2} d^{4} e^{2} + 3 \, a^{6} c d^{2} e^{4} + a^{7} e^{6} + {\left (a^{2} c^{5} d^{6} + 3 \, a^{3} c^{4} d^{4} e^{2} + 3 \, a^{4} c^{3} d^{2} e^{4} + a^{5} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} + 3 \, a^{4} c^{3} d^{4} e^{2} + 3 \, a^{5} c^{2} d^{2} e^{4} + a^{6} c e^{6}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 935, normalized size = 6.07 \begin {gather*} -\frac {2 \, \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^{4}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {{\left ({\left (\frac {{\left (2 \, c^{10} d^{15} + 17 \, a c^{9} d^{13} e^{2} + 60 \, a^{2} c^{8} d^{11} e^{4} + 115 \, a^{3} c^{7} d^{9} e^{6} + 130 \, a^{4} c^{6} d^{7} e^{8} + 87 \, a^{5} c^{5} d^{5} e^{10} + 32 \, a^{6} c^{4} d^{3} e^{12} + 5 \, a^{7} c^{3} d e^{14}\right )} x}{a^{2} c^{9} d^{16} + 8 \, a^{3} c^{8} d^{14} e^{2} + 28 \, a^{4} c^{7} d^{12} e^{4} + 56 \, a^{5} c^{6} d^{10} e^{6} + 70 \, a^{6} c^{5} d^{8} e^{8} + 56 \, a^{7} c^{4} d^{6} e^{10} + 28 \, a^{8} c^{3} d^{4} e^{12} + 8 \, a^{9} c^{2} d^{2} e^{14} + a^{10} c e^{16}} + \frac {3 \, {\left (a^{2} c^{8} d^{12} e^{3} + 6 \, a^{3} c^{7} d^{10} e^{5} + 15 \, a^{4} c^{6} d^{8} e^{7} + 20 \, a^{5} c^{5} d^{6} e^{9} + 15 \, a^{6} c^{4} d^{4} e^{11} + 6 \, a^{7} c^{3} d^{2} e^{13} + a^{8} c^{2} e^{15}\right )}}{a^{2} c^{9} d^{16} + 8 \, a^{3} c^{8} d^{14} e^{2} + 28 \, a^{4} c^{7} d^{12} e^{4} + 56 \, a^{5} c^{6} d^{10} e^{6} + 70 \, a^{6} c^{5} d^{8} e^{8} + 56 \, a^{7} c^{4} d^{6} e^{10} + 28 \, a^{8} c^{3} d^{4} e^{12} + 8 \, a^{9} c^{2} d^{2} e^{14} + a^{10} c e^{16}}\right )} x + \frac {3 \, {\left (a c^{9} d^{15} + 8 \, a^{2} c^{8} d^{13} e^{2} + 27 \, a^{3} c^{7} d^{11} e^{4} + 50 \, a^{4} c^{6} d^{9} e^{6} + 55 \, a^{5} c^{5} d^{7} e^{8} + 36 \, a^{6} c^{4} d^{5} e^{10} + 13 \, a^{7} c^{3} d^{3} e^{12} + 2 \, a^{8} c^{2} d e^{14}\right )}}{a^{2} c^{9} d^{16} + 8 \, a^{3} c^{8} d^{14} e^{2} + 28 \, a^{4} c^{7} d^{12} e^{4} + 56 \, a^{5} c^{6} d^{10} e^{6} + 70 \, a^{6} c^{5} d^{8} e^{8} + 56 \, a^{7} c^{4} d^{6} e^{10} + 28 \, a^{8} c^{3} d^{4} e^{12} + 8 \, a^{9} c^{2} d^{2} e^{14} + a^{10} c e^{16}}\right )} x + \frac {a^{2} c^{8} d^{14} e + 10 \, a^{3} c^{7} d^{12} e^{3} + 39 \, a^{4} c^{6} d^{10} e^{5} + 80 \, a^{5} c^{5} d^{8} e^{7} + 95 \, a^{6} c^{4} d^{6} e^{9} + 66 \, a^{7} c^{3} d^{4} e^{11} + 25 \, a^{8} c^{2} d^{2} e^{13} + 4 \, a^{9} c e^{15}}{a^{2} c^{9} d^{16} + 8 \, a^{3} c^{8} d^{14} e^{2} + 28 \, a^{4} c^{7} d^{12} e^{4} + 56 \, a^{5} c^{6} d^{10} e^{6} + 70 \, a^{6} c^{5} d^{8} e^{8} + 56 \, a^{7} c^{4} d^{6} e^{10} + 28 \, a^{8} c^{3} d^{4} e^{12} + 8 \, a^{9} c^{2} d^{2} e^{14} + a^{10} c e^{16}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 454, normalized size = 2.95 \begin {gather*} \frac {c d \,e^{2} x}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a}-\frac {e^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {e^{3}}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {c d x}{3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}} a}+\frac {2 c d x}{3 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a^{2}}+\frac {e}{3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.79, size = 283, normalized size = 1.84 \begin {gather*} \frac {c d x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2}\right )}} + \frac {c d x}{2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} + \frac {\sqrt {c x^{2} + a} a c^{2} d^{4}}{e^{2}} + \sqrt {c x^{2} + a} a^{3} e^{2}} + \frac {2 \, c d x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c d^{2} + \sqrt {c x^{2} + a} a^{3} e^{2}\right )}} + \frac {1}{\frac {\sqrt {c x^{2} + a} c^{2} d^{4}}{e^{3}} + \frac {2 \, \sqrt {c x^{2} + a} a c d^{2}}{e} + \sqrt {c x^{2} + a} a^{2} e} + \frac {1}{3 \, {\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2}}{e} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e\right )}} + \frac {\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 )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,\left (d+e\,x\right )} \,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 {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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