Optimal. Leaf size=244 \[ \frac {e \sqrt {a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 244, 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, 807, 725, 206} \[ \frac {e \sqrt {a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 741
Rule 807
Rule 823
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 \left (c d^2+2 a e^2\right )-3 c d e x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {\int \frac {-2 a c e^2 \left (c d^2-4 a e^2\right )+c^2 d e \left (2 c d^2+7 a e^2\right ) x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (5 c d e^4\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (5 c d e^4\right ) \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 )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {5 c d 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 )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 237, normalized size = 0.97 \[ \frac {-3 a^4 e^5+2 a^3 c e^3 \left (7 d^2+4 d e x-6 e^2 x^2\right )+a^2 c^2 e \left (2 d^4+11 d^3 e x+21 d^2 e^2 x^2+7 d e^3 x^3-8 e^4 x^4\right )+3 a c^3 d^2 x \left (d^3+d^2 e x+3 d e^2 x^2+3 e^3 x^3\right )+2 c^4 d^4 x^3 (d+e x)}{3 a^2 \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )^3}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.83, size = 1678, normalized size = 6.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.95, size = 1073, normalized size = 4.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 667, normalized size = 2.73 \[ \frac {5 c^{2} d^{2} e^{2} x}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \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 {5 c d \,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 )^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {5 c d \,e^{3}}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \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 {5 c^{2} d^{2} x}{3 \left (a \,e^{2}+c \,d^{2}\right )^{2} \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 {10 c^{2} d^{2} x}{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}}}\, a^{2}}+\frac {5 c d e}{3 \left (a \,e^{2}+c \,d^{2}\right )^{2} \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}}}-\frac {4 c 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 {8 c 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 {1}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\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}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.15, size = 563, normalized size = 2.31 \[ \frac {5 \, c^{2} d^{2} x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d^{4} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d^{2} e^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{4}\right )}} + \frac {5 \, c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + \frac {\sqrt {c x^{2} + a} a c^{3} d^{6}}{e^{2}} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}} + \frac {10 \, c^{2} d^{2} x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}\right )}} + \frac {5 \, c d}{\frac {\sqrt {c x^{2} + a} c^{3} d^{6}}{e^{3}} + \frac {3 \, \sqrt {c x^{2} + a} a c^{2} d^{4}}{e} + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e + \sqrt {c x^{2} + a} a^{3} e^{3}} + \frac {5 \, c d}{3 \, {\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{4}}{e} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} e + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{3}\right )}} - \frac {4 \, c x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2}\right )}} - \frac {8 \, c x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c d^{2} + \sqrt {c x^{2} + a} a^{3} e^{2}\right )}} - \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2} x + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{3}}{e} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d e} + \frac {5 \, c d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{3}} \]
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 \[ \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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