Optimal. Leaf size=144 \[ \frac {a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac {d x \sqrt {a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac {e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]
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Rubi [A] time = 0.11, antiderivative size = 144, 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 = {743, 780, 195, 217, 206} \begin {gather*} \frac {e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac {a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {d x \sqrt {a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac {e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 743
Rule 780
Rubi steps
\begin {align*} \int (d+e x)^3 \sqrt {a+c x^2} \, dx &=\frac {e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {\int (d+e x) \left (5 c d^2-2 a e^2+7 c d e x\right ) \sqrt {a+c x^2} \, dx}{5 c}\\ &=\frac {e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac {\left (d \left (4 c d^2-3 a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{4 c}\\ &=\frac {d \left (4 c d^2-3 a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac {\left (a d \left (4 c d^2-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c}\\ &=\frac {d \left (4 c d^2-3 a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac {\left (a d \left (4 c d^2-3 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 )}{8 c}\\ &=\frac {d \left (4 c d^2-3 a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac {a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 132, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+15 a \sqrt {c} d \left (4 c d^2-3 a e^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{120 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 146, normalized size = 1.01 \begin {gather*} \frac {\left (3 a^2 d e^2-4 a c d^3\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (-16 a^2 e^3+120 a c d^2 e+45 a c d e^2 x+8 a c e^3 x^2+60 c^2 d^3 x+120 c^2 d^2 e x^2+90 c^2 d e^2 x^3+24 c^2 e^3 x^4\right )}{120 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 286, normalized size = 1.99 \begin {gather*} \left [-\frac {15 \, {\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \, {\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{2}}, -\frac {15 \, {\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \, {\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{120 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 144, normalized size = 1.00 \begin {gather*} \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (4 \, x e^{3} + 15 \, d e^{2}\right )} x + \frac {4 \, {\left (15 \, c^{3} d^{2} e + a c^{2} e^{3}\right )}}{c^{3}}\right )} x + \frac {15 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}}{c^{3}}\right )} x + \frac {8 \, {\left (15 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}\right )}}{c^{3}}\right )} - \frac {{\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 164, normalized size = 1.14 \begin {gather*} -\frac {3 a^{2} d \,e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}+\frac {a \,d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {3 \sqrt {c \,x^{2}+a}\, a d \,e^{2} x}{8 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e^{3} x^{2}}{5 c}+\frac {\sqrt {c \,x^{2}+a}\, d^{3} x}{2}+\frac {3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} d \,e^{2} x}{4 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,e^{3}}{15 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d^{2} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 149, normalized size = 1.03 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e^{3} x^{2}}{5 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + a} d^{3} x + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d e^{2} x}{4 \, c} - \frac {3 \, \sqrt {c x^{2} + a} a d e^{2} x}{8 \, c} + \frac {a d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} - \frac {3 \, a^{2} d e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2} e}{c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{3}}{15 \, c^{2}} \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 \sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.08, size = 265, normalized size = 1.84 \begin {gather*} \frac {3 a^{\frac {3}{2}} d e^{2} x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {\sqrt {a} d^{3} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {9 \sqrt {a} d e^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 a^{2} d e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} + \frac {a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 \sqrt {c}} + 3 d^{2} 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 ) + e^{3} \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 ) + \frac {3 c d e^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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