Optimal. Leaf size=189 \[ \frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}+\frac {5 a^2 x \sqrt {a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac {x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac {5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]
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Rubi [A] time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {743, 641, 195, 217, 206} \[ \frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}+\frac {5 a^2 x \sqrt {a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac {x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac {5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rule 743
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (8 c d^2-a e^2+9 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (8 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a \left (8 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^2 \left (8 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{128 c}\\ &=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 162, normalized size = 0.86 \[ \frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^3 e (256 d+35 e x)+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )-105 a^3 \left (a e^2-8 c d^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{2688 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 380, normalized size = 2.01 \[ \left [\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{5376 \, c^{2}}, -\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2688 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 190, normalized size = 1.01 \[ \frac {1}{2688} \, {\left (\frac {768 \, a^{3} d e}{c} + {\left (2 \, {\left (1152 \, a^{2} d e + {\left (4 \, {\left (288 \, a c d e + {\left (6 \, {\left (7 \, c^{2} x e^{2} + 16 \, c^{2} d e\right )} x + \frac {7 \, {\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {7 \, {\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {21 \, {\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt {c x^{2} + a} - \frac {5 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 200, normalized size = 1.06 \[ -\frac {5 a^{4} e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}+\frac {5 a^{3} d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}-\frac {5 \sqrt {c \,x^{2}+a}\, a^{3} e^{2} x}{128 c}+\frac {5 \sqrt {c \,x^{2}+a}\, a^{2} d^{2} x}{16}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} e^{2} x}{192 c}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,d^{2} x}{24}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} a \,e^{2} x}{48 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} d^{2} x}{6}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} e^{2} x}{8 c}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d e}{7 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 185, normalized size = 0.98 \[ \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{2} x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{2} x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} a^{3} e^{2} x}{128 \, c} + \frac {5 \, a^{3} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {5 \, a^{4} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e}{7 \, c} \]
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 \[ \int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.73, size = 539, normalized size = 2.85 \[ \frac {5 a^{\frac {7}{2}} e^{2} x}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} d^{2} x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} e^{2} x^{3}}{384 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} c d^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} c e^{2} x^{5}}{192 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 \sqrt {a} c^{2} d^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {23 \sqrt {a} c^{2} e^{2} x^{7}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {5 a^{4} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {3}{2}}} + \frac {5 a^{3} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + 2 a^{2} d e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 4 a c d e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 c^{2} d e \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {c^{3} d^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{3} e^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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