Optimal. Leaf size=266 \[ \frac {2 b d^2 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {272, 45, 5804,
12, 380} \begin {gather*} \frac {\left (c^2 d x^2+d\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {c^2 d x^2+d}}{21 \sqrt {c^2 x^2+1}}-\frac {b d^2 x^3 \sqrt {c^2 d x^2+d}}{189 c \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^9 \sqrt {c^2 d x^2+d}}{81 \sqrt {c^2 x^2+1}}+\frac {2 b d^2 x \sqrt {c^2 d x^2+d}}{63 c^3 \sqrt {c^2 x^2+1}}-\frac {19 b c^3 d^2 x^7 \sqrt {c^2 d x^2+d}}{441 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 5804
Rubi steps
\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt {1+c^2 x^2}}+\left (a+b \sinh ^{-1}(c x)\right ) \int x^3 \left (d+c^2 d x^2\right )^{5/2} \, dx\\ &=-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}+\frac {1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Subst}\left (\int x \left (d+c^2 d x\right )^{5/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}+\frac {1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Subst}\left (\int \left (-\frac {\left (d+c^2 d x\right )^{5/2}}{c^2}+\frac {\left (d+c^2 d x\right )^{7/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d^2 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 140, normalized size = 0.53 \begin {gather*} \frac {d^2 \sqrt {d+c^2 d x^2} \left (63 a \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right )-b c x \sqrt {1+c^2 x^2} \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+63 b \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right ) \sinh ^{-1}(c x)\right )}{3969 c^4 \left (1+c^2 x^2\right )} \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. \(995\) vs.
\(2(228)=456\).
time = 1.58, size = 996, normalized size = 3.74
method | result | size |
default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 x^{10} c^{10}+256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 x^{8} c^{8}+576 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+688 x^{6} c^{6}+432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}+120 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+41 c^{2} x^{2}+9 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+9 \arcsinh \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 x^{8} c^{8}+64 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+144 x^{6} c^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+7 \arcsinh \left (c x \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+3 \arcsinh \left (c x \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (\arcsinh \left (c x \right )-1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+\arcsinh \left (c x \right )\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+3 \arcsinh \left (c x \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 x^{8} c^{8}-64 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+144 x^{6} c^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+7 \arcsinh \left (c x \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 x^{10} c^{10}-256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 x^{8} c^{8}-576 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+688 x^{6} c^{6}-432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}-120 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+41 c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+9 \arcsinh \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(996\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 156, normalized size = 0.59 \begin {gather*} \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} + 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} + 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 263, normalized size = 0.99 \begin {gather*} \frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} + 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} + 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} - 2 \, b d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (441 \, a c^{10} d^{2} x^{10} + 1638 \, a c^{8} d^{2} x^{8} + 2142 \, a c^{6} d^{2} x^{6} + 1008 \, a c^{4} d^{2} x^{4} - 63 \, a c^{2} d^{2} x^{2} - 126 \, a d^{2} - {\left (49 \, b c^{9} d^{2} x^{9} + 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} + 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} + c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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