Optimal. Leaf size=211 \[ \frac {\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5}-\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \]
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Rubi [A] time = 0.39, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1654, 815, 844, 217, 206, 725} \[ \frac {\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}-\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^4}+\frac {d^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 815
Rule 844
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx &=\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\sqrt {a+c x^2} \left (-a d e^2-e \left (3 c d^2+a e^2\right ) x-7 c d e^2 x^2\right )}{d+e x} \, dx}{4 c e^3}\\ &=-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\left (-3 a c d e^4+3 c e^3 \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{12 c^2 e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {-3 a c^2 d e^4 \left (4 c d^2+a e^2\right )+3 c^2 e^3 \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{24 c^3 e^7}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (d^3 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (d^3 \left (c d^2+a e^2\right )\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 )}{e^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c e^5}\\ &=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 225, normalized size = 1.07 \[ \frac {24 c^{3/2} d^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+24 c d^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+e \sqrt {a+c x^2} \left (a e^2 (3 e x-8 d)+c \left (-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )\right )}{24 c e^5}-\frac {\sqrt {a} \sqrt {a+c x^2} \left (a e^2-4 c d^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 c^{3/2} e^3 \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 13.97, size = 963, normalized size = 4.56 \[ \left [\frac {24 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{2} e^{5}}, \frac {48 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{2} e^{5}}, \frac {12 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{2} e^{5}}, \frac {24 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{2} e^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 201, normalized size = 0.95 \[ -\frac {2 \, {\left (c d^{5} + a d^{3} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-5\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x + \frac {3 \, {\left (4 \, c^{2} d^{2} e^{12} + a c e^{14}\right )} e^{\left (-15\right )}}{c^{2}}\right )} x - \frac {8 \, {\left (3 \, c^{2} d^{3} e^{11} + a c d e^{13}\right )} e^{\left (-15\right )}}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 515, normalized size = 2.44 \[ \frac {a \,d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{4}}+\frac {c \,d^{5} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{6}}-\frac {a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}} e}+\frac {a \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}\, e^{3}}+\frac {\sqrt {c}\, d^{4} \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e^{5}}-\frac {\sqrt {c \,x^{2}+a}\, a x}{8 c e}+\frac {\sqrt {c \,x^{2}+a}\, d^{2} x}{2 e^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} x}{4 c e}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{3}}{e^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d}{3 c \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 207, normalized size = 0.98 \[ \frac {\sqrt {c x^{2} + a} d^{2} x}{2 \, e^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, c e} - \frac {\sqrt {c x^{2} + a} a x}{8 \, c e} + \frac {\sqrt {c} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{5}} + \frac {a d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} e^{3}} - \frac {a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}} e} - \frac {\sqrt {a + \frac {c d^{2}}{e^{2}}} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{e^{4}} - \frac {\sqrt {c x^{2} + a} d^{3}}{e^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d}{3 \, c e^{2}} \]
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 \[ \int \frac {x^3\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {a + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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