Optimal. Leaf size=110 \[ \frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {757, 794, 223,
212} \begin {gather*} \frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}+\frac {e \sqrt {a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^2}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 757
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {\int \frac {(d+e x) \left (3 c d^2-2 a e^2+5 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{3 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 91, normalized size = 0.83 \begin {gather*} \frac {e \sqrt {a+c x^2} \left (-4 a e^2+c \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 \sqrt {c} d \left (-2 c d^2+3 a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 124, normalized size = 1.13
method | result | size |
risch | \(-\frac {e \left (-2 c \,e^{2} x^{2}-9 c d e x +4 e^{2} a -18 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{2}}-\frac {3 d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) e^{2} a}{2 c^{\frac {3}{2}}}+\frac {d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(94\) |
default | \(e^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}\right )+3 d \,e^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {3 d^{2} e \sqrt {c \,x^{2}+a}}{c}+\frac {d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 108, normalized size = 0.98 \begin {gather*} \frac {d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} x^{2} e^{3}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + a} d x e^{2}}{2 \, c} - \frac {3 \, a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{2 \, c^{\frac {3}{2}}} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} e}{c} - \frac {2 \, \sqrt {c x^{2} + a} a e^{3}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 174, normalized size = 1.58 \begin {gather*} \left [-\frac {3 \, {\left (2 \, c d^{3} - 3 \, a 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 (9 \, c d x e^{2} + 18 \, c d^{2} e + 2 \, {\left (c x^{2} - 2 \, a\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, c^{2}}, -\frac {3 \, {\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (9 \, c d x e^{2} + 18 \, c d^{2} e + 2 \, {\left (c x^{2} - 2 \, a\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.27, size = 216, normalized size = 1.96 \begin {gather*} \frac {3 \sqrt {a} d e^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2 c} - \frac {3 a d e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 c^{\frac {3}{2}}} + d^{3} \left (\begin {cases} \frac {\sqrt {- \frac {a}{c}} \operatorname {asin}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c < 0 \\\frac {\sqrt {\frac {a}{c}} \operatorname {asinh}{\left (x \sqrt {\frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c > 0 \\\frac {\sqrt {- \frac {a}{c}} \operatorname {acosh}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {- a}} & \text {for}\: c > 0 \wedge a < 0 \end {cases}\right ) + 3 d^{2} e \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: c = 0 \\\frac {\sqrt {a + c x^{2}}}{c} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} - \frac {2 a \sqrt {a + c x^{2}}}{3 c^{2}} + \frac {x^{2} \sqrt {a + c x^{2}}}{3 c} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 90, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, x e^{3}}{c} + \frac {9 \, d e^{2}}{c}\right )} + \frac {2 \, {\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac {{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {3}{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 \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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