Optimal. Leaf size=405 \[ -\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.46, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5808, 5806,
5812, 5783, 5776, 327, 221, 14, 5803, 12, 470} \begin {gather*} -\frac {b d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {c^2 x^2+1}}+\frac {d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}-\frac {7 b c d x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^6 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {c^2 x^2+1}}-\frac {7 b^2 d x \sqrt {c^2 d x^2+d}}{1152 c^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {c^2 d x^2+d}+\frac {43 b^2 d x^3 \sqrt {c^2 d x^2+d}}{1728}+\frac {7 b^2 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 221
Rule 327
Rule 470
Rule 5776
Rule 5783
Rule 5803
Rule 5806
Rule 5808
Rule 5812
Rubi steps
\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \int x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{64} b^2 d x^3 \sqrt {d+c^2 d x^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{27 \sqrt {1+c^2 x^2}}\\ &=\frac {b^2 d x \sqrt {d+c^2 d x^2}}{128 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{128 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{72 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 508, normalized size = 1.25 \begin {gather*} \frac {864 a^2 c d x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+4032 a^2 c^3 d x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 d x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-288 b^2 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^3+216 a b d \sqrt {d+c^2 d x^2} \cosh \left (2 \sinh ^{-1}(c x)\right )-108 a b d \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b d \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )-864 a^2 d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-108 b^2 d \sqrt {d+c^2 d x^2} \sinh \left (2 \sinh ^{-1}(c x)\right )+27 b^2 d \sqrt {d+c^2 d x^2} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 d \sqrt {d+c^2 d x^2} \sinh \left (6 \sinh ^{-1}(c x)\right )+12 b d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (18 b \cosh \left (2 \sinh ^{-1}(c x)\right )-9 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )-36 a \sinh \left (2 \sinh ^{-1}(c x)\right )+36 a \sinh \left (4 \sinh ^{-1}(c x)\right )+12 a \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+72 b d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2 \left (-12 a-3 b \sinh \left (2 \sinh ^{-1}(c x)\right )+3 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )}{13824 c^3 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1551\) vs.
\(2(351)=702\).
time = 2.70, size = 1552, normalized size = 3.83
method | result | size |
default | \(\text {Expression too large to display}\) | \(1552\) |
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: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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