Optimal. Leaf size=266 \[ -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {1+c^2 x^2}}+\frac {7 b c^3 d^2 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5807, 5785,
5783, 30, 14, 272, 45} \begin {gather*} -\frac {5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x}-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {5}{2} c^4 d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {c^2 x^2+1}}-\frac {b c d^2 \sqrt {c^2 d x^2+d}}{6 x^2 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}+\frac {7 b c^3 d^2 \log (x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5785
Rule 5807
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {1}{3} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1+c^2 x\right )^2}{x^2} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (c^4+\frac {1}{x^2}+\frac {2 c^2}{x}\right ) \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {1}{x}+c^2 x\right ) \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^5 d^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {1+c^2 x^2}}+\frac {7 b c^3 d^2 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 286, normalized size = 1.08 \begin {gather*} \frac {d^2 \left (4 a \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (-2-14 c^2 x^2+3 c^4 x^4\right )+24 b c^2 x^2 \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+c x \sinh ^{-1}(c x)^2+2 c x \log (c x)\right )-4 b \sqrt {d+c^2 d x^2} \left (c x+2 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)-2 c^3 x^3 \log (c x)\right )+60 a c^3 \sqrt {d} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-3 b c^3 x^3 \sqrt {d+c^2 d x^2} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )\right )}{24 x^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. \(1315\) vs.
\(2(230)=460\).
time = 3.70, size = 1316, normalized size = 4.95
method | result | size |
default | \(\text {Expression too large to display}\) | \(1316\) |
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 \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{4}}\, dx \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 \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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