Optimal. Leaf size=283 \[ -\frac {5 b e \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{24 d^2 \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{128 d^{7/2} \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt {c+d x^2+e x} \left (-16 a d^2+4 b c d-5 b e^2\right )}{64 d^3 \left (a+b x^2\right )}+\frac {b x \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{4 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.34, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6744, 1661, 640, 612, 621, 206} \begin {gather*} -\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt {c+d x^2+e x} \left (-16 a d^2+4 b c d-5 b e^2\right )}{64 d^3 \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{128 d^{7/2} \left (a+b x^2\right )}-\frac {5 b e \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{24 d^2 \left (a+b x^2\right )}+\frac {b x \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{4 d \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rule 6744
Rubi steps
\begin {align*} \int \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (2 a b+2 b^2 x^2\right ) \sqrt {c+e x+d x^2} \, dx}{2 a b+2 b^2 x^2}\\ &=\frac {b x \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 d \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (-2 b (b c-4 a d)-5 b^2 e x\right ) \sqrt {c+e x+d x^2} \, dx}{4 d \left (2 a b+2 b^2 x^2\right )}\\ &=-\frac {5 b e \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 d^2 \left (a+b x^2\right )}+\frac {b x \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 d \left (a+b x^2\right )}+\frac {\left (\left (-4 b d (b c-4 a d)+5 b^2 e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \sqrt {c+e x+d x^2} \, dx}{8 d^2 \left (2 a b+2 b^2 x^2\right )}\\ &=-\frac {\left (4 b c d-16 a d^2-5 b e^2\right ) (e+2 d x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{64 d^3 \left (a+b x^2\right )}-\frac {5 b e \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 d^2 \left (a+b x^2\right )}+\frac {b x \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 d \left (a+b x^2\right )}+\frac {\left (\left (4 c d-e^2\right ) \left (-4 b d (b c-4 a d)+5 b^2 e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{64 d^3 \left (2 a b+2 b^2 x^2\right )}\\ &=-\frac {\left (4 b c d-16 a d^2-5 b e^2\right ) (e+2 d x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{64 d^3 \left (a+b x^2\right )}-\frac {5 b e \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 d^2 \left (a+b x^2\right )}+\frac {b x \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 d \left (a+b x^2\right )}+\frac {\left (\left (4 c d-e^2\right ) \left (-4 b d (b c-4 a d)+5 b^2 e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{32 d^3 \left (2 a b+2 b^2 x^2\right )}\\ &=-\frac {\left (4 b c d-16 a d^2-5 b e^2\right ) (e+2 d x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{64 d^3 \left (a+b x^2\right )}-\frac {5 b e \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 d^2 \left (a+b x^2\right )}+\frac {b x \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 d \left (a+b x^2\right )}-\frac {\left (4 c d-e^2\right ) \left (4 b c d-16 a d^2-5 b e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{128 d^{7/2} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 168, normalized size = 0.59 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (2 \sqrt {d} \sqrt {c+x (d x+e)} \left (48 a d^2 (2 d x+e)+b \left (4 c d (6 d x-13 e)+48 d^3 x^3+8 d^2 e x^2-10 d e^2 x+15 e^3\right )\right )-3 \left (4 c d-e^2\right ) \left (-16 a d^2+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )\right )}{384 d^{7/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 184, normalized size = 0.65 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (\frac {\left (-64 a c d^3+16 a d^2 e^2+16 b c^2 d^2-24 b c d e^2+5 b e^4\right ) \log \left (-2 \sqrt {d} \sqrt {c+d x^2+e x}+2 d x+e\right )}{128 d^{7/2}}+\frac {\sqrt {c+d x^2+e x} \left (96 a d^3 x+48 a d^2 e+24 b c d^2 x-52 b c d e+48 b d^3 x^3+8 b d^2 e x^2-10 b d e^2 x+15 b e^3\right )}{192 d^3}\right )}{a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.49, size = 361, normalized size = 1.28 \begin {gather*} \left [\frac {3 \, {\left (16 \, b c^{2} d^{2} - 64 \, a c d^{3} + 5 \, b e^{4} - 8 \, {\left (3 \, b c d - 2 \, a d^{2}\right )} e^{2}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (48 \, b d^{4} x^{3} + 8 \, b d^{3} e x^{2} + 15 \, b d e^{3} - 4 \, {\left (13 \, b c d^{2} - 12 \, a d^{3}\right )} e + 2 \, {\left (12 \, b c d^{3} + 48 \, a d^{4} - 5 \, b d^{2} e^{2}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{768 \, d^{4}}, \frac {3 \, {\left (16 \, b c^{2} d^{2} - 64 \, a c d^{3} + 5 \, b e^{4} - 8 \, {\left (3 \, b c d - 2 \, a d^{2}\right )} e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (48 \, b d^{4} x^{3} + 8 \, b d^{3} e x^{2} + 15 \, b d e^{3} - 4 \, {\left (13 \, b c d^{2} - 12 \, a d^{3}\right )} e + 2 \, {\left (12 \, b c d^{3} + 48 \, a d^{4} - 5 \, b d^{2} e^{2}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{384 \, d^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 265, normalized size = 0.94 \begin {gather*} \frac {1}{192} \, \sqrt {d x^{2} + x e + c} {\left (2 \, {\left (4 \, {\left (6 \, b x \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {b e \mathrm {sgn}\left (b x^{2} + a\right )}{d}\right )} x + \frac {12 \, b c d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, a d^{3} \mathrm {sgn}\left (b x^{2} + a\right ) - 5 \, b d e^{2} \mathrm {sgn}\left (b x^{2} + a\right )}{d^{3}}\right )} x - \frac {52 \, b c d e \mathrm {sgn}\left (b x^{2} + a\right ) - 48 \, a d^{2} e \mathrm {sgn}\left (b x^{2} + a\right ) - 15 \, b e^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{d^{3}}\right )} + \frac {{\left (16 \, b c^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) - 64 \, a c d^{3} \mathrm {sgn}\left (b x^{2} + a\right ) - 24 \, b c d e^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 16 \, a d^{2} e^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 5 \, b e^{4} \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} \sqrt {d} - e \right |}\right )}{128 \, d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 373, normalized size = 1.32 \begin {gather*} \frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (192 a c \,d^{4} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-48 a \,d^{3} e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-48 b \,c^{2} d^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+72 b c \,d^{2} e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-15 b d \,e^{4} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+192 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {9}{2}} x -48 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {7}{2}} x +60 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {5}{2}} e^{2} x +96 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {7}{2}} e -24 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {5}{2}} e +96 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {7}{2}} x +30 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {3}{2}} e^{3}-80 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {5}{2}} e \right )}{384 \left (b \,x^{2}+a \right ) d^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.00 \begin {gather*} \int \sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {d\,x^2+e\,x+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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