Optimal. Leaf size=161 \[ \frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 161, 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 = {757, 847, 794,
223, 212} \begin {gather*} \frac {\left (3 a^2 e^4-24 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^{5/2}}+\frac {e \sqrt {a+c x^2} \left (e x \left (26 c d^2-9 a e^2\right )+4 d \left (19 c d^2-16 a e^2\right )\right )}{24 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^3}{4 c}+\frac {7 d e \sqrt {a+c x^2} (d+e x)^2}{12 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 757
Rule 794
Rule 847
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\sqrt {a+c x^2}} \, dx &=\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (4 c d^2-3 a e^2+7 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{4 c}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (c d \left (12 c d^2-23 a e^2\right )+c e \left (26 c d^2-9 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{12 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 127, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a+c x^2} \left (96 c d^3 e-64 a d e^3+72 c d^2 e^2 x-9 a e^4 x+32 c d e^3 x^2+6 c e^4 x^3\right )}{24 c^2}+\frac {\left (-8 c^2 d^4+24 a c d^2 e^2-3 a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{8 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 194, normalized size = 1.20
method | result | size |
risch | \(-\frac {e \left (-6 e^{3} c \,x^{3}-32 c d \,e^{2} x^{2}+9 a \,e^{3} x -72 c \,d^{2} e x +64 a d \,e^{2}-96 c \,d^{3}\right ) \sqrt {c \,x^{2}+a}}{24 c^{2}}+\frac {3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) a^{2} e^{4}}{8 c^{\frac {5}{2}}}-\frac {3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) a \,d^{2} e^{2}}{c^{\frac {3}{2}}}+\frac {d^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(143\) |
default | \(e^{4} \left (\frac {x^{3} \sqrt {c \,x^{2}+a}}{4 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}\right )+6 d^{2} e^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {4 d^{3} e \sqrt {c \,x^{2}+a}}{c}+\frac {d^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 170, normalized size = 1.06 \begin {gather*} \frac {d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} x^{3} e^{4}}{4 \, c} + \frac {4 \, \sqrt {c x^{2} + a} d x^{2} e^{3}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} x e^{2}}{c} - \frac {3 \, a d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{c^{\frac {3}{2}}} + \frac {4 \, \sqrt {c x^{2} + a} d^{3} e}{c} - \frac {3 \, \sqrt {c x^{2} + a} a x e^{4}}{8 \, c^{2}} + \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{4}}{8 \, c^{\frac {5}{2}}} - \frac {8 \, \sqrt {c x^{2} + a} a d e^{3}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 256, normalized size = 1.59 \begin {gather*} \left [\frac {3 \, {\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, 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 (72 \, c^{2} d^{2} x e^{2} + 96 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} - 3 \, a c x\right )} e^{4} + 32 \, {\left (c^{2} d x^{2} - 2 \, a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{48 \, c^{3}}, -\frac {3 \, {\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (72 \, c^{2} d^{2} x e^{2} + 96 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} - 3 \, a c x\right )} e^{4} + 32 \, {\left (c^{2} d x^{2} - 2 \, a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{24 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.71, size = 330, normalized size = 2.05 \begin {gather*} - \frac {3 a^{\frac {3}{2}} e^{4} x}{8 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} d^{2} e^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{c} - \frac {\sqrt {a} e^{4} x^{3}}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{2} e^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {5}{2}}} - \frac {3 a d^{2} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{c^{\frac {3}{2}}} + d^{4} \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 ) + 4 d^{3} 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 ) + 4 d 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 ) + \frac {e^{4} 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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Giac [A]
time = 1.56, size = 133, normalized size = 0.83 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, x {\left (\frac {3 \, x e^{4}}{c} + \frac {16 \, d e^{3}}{c}\right )} + \frac {9 \, {\left (8 \, c^{3} d^{2} e^{2} - a c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac {32 \, {\left (3 \, c^{3} d^{3} e - 2 \, a c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac {{\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {5}{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 )}^4}{\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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