Optimal. Leaf size=130 \[ -\frac {b p}{15 a x^{5/2}}+\frac {b^2 p}{12 a^2 x^2}-\frac {b^3 p}{9 a^3 x^{3/2}}+\frac {b^4 p}{6 a^4 x}-\frac {b^5 p}{3 a^5 \sqrt {x}}+\frac {b^6 p \log \left (a+b \sqrt {x}\right )}{3 a^6}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{3 x^3}-\frac {b^6 p \log (x)}{6 a^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 2442, 46}
\begin {gather*} \frac {b^6 p \log \left (a+b \sqrt {x}\right )}{3 a^6}-\frac {b^6 p \log (x)}{6 a^6}-\frac {b^5 p}{3 a^5 \sqrt {x}}+\frac {b^4 p}{6 a^4 x}-\frac {b^3 p}{9 a^3 x^{3/2}}+\frac {b^2 p}{12 a^2 x^2}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{3 x^3}-\frac {b p}{15 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^4} \, dx &=2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^7} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \frac {1}{x^6 (a+b x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \left (\frac {1}{a x^6}-\frac {b}{a^2 x^5}+\frac {b^2}{a^3 x^4}-\frac {b^3}{a^4 x^3}+\frac {b^4}{a^5 x^2}-\frac {b^5}{a^6 x}+\frac {b^6}{a^6 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {b p}{15 a x^{5/2}}+\frac {b^2 p}{12 a^2 x^2}-\frac {b^3 p}{9 a^3 x^{3/2}}+\frac {b^4 p}{6 a^4 x}-\frac {b^5 p}{3 a^5 \sqrt {x}}+\frac {b^6 p \log \left (a+b \sqrt {x}\right )}{3 a^6}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{3 x^3}-\frac {b^6 p \log (x)}{6 a^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 114, normalized size = 0.88 \begin {gather*} \frac {a b p \sqrt {x} \left (-12 a^4+15 a^3 b \sqrt {x}-20 a^2 b^2 x+30 a b^3 x^{3/2}-60 b^4 x^2\right )+60 b^6 p x^3 \log \left (a+b \sqrt {x}\right )-60 a^6 \log \left (c \left (a+b \sqrt {x}\right )^p\right )-30 b^6 p x^3 \log (x)}{180 a^6 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 98, normalized size = 0.75 \begin {gather*} \frac {1}{180} \, b p {\left (\frac {60 \, b^{5} \log \left (b \sqrt {x} + a\right )}{a^{6}} - \frac {30 \, b^{5} \log \left (x\right )}{a^{6}} - \frac {60 \, b^{4} x^{2} - 30 \, a b^{3} x^{\frac {3}{2}} + 20 \, a^{2} b^{2} x - 15 \, a^{3} b \sqrt {x} + 12 \, a^{4}}{a^{5} x^{\frac {5}{2}}}\right )} - \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 109, normalized size = 0.84 \begin {gather*} -\frac {60 \, b^{6} p x^{3} \log \left (\sqrt {x}\right ) - 30 \, a^{2} b^{4} p x^{2} - 15 \, a^{4} b^{2} p x + 60 \, a^{6} \log \left (c\right ) - 60 \, {\left (b^{6} p x^{3} - a^{6} p\right )} \log \left (b \sqrt {x} + a\right ) + 4 \, {\left (15 \, a b^{5} p x^{2} + 5 \, a^{3} b^{3} p x + 3 \, a^{5} b p\right )} \sqrt {x}}{180 \, a^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 324 vs.
\(2 (104) = 208\).
time = 4.52, size = 324, normalized size = 2.49 \begin {gather*} -\frac {\frac {60 \, b^{7} p \log \left (b \sqrt {x} + a\right )}{{\left (b \sqrt {x} + a\right )}^{6} - 6 \, {\left (b \sqrt {x} + a\right )}^{5} a + 15 \, {\left (b \sqrt {x} + a\right )}^{4} a^{2} - 20 \, {\left (b \sqrt {x} + a\right )}^{3} a^{3} + 15 \, {\left (b \sqrt {x} + a\right )}^{2} a^{4} - 6 \, {\left (b \sqrt {x} + a\right )} a^{5} + a^{6}} - \frac {60 \, b^{7} p \log \left (b \sqrt {x} + a\right )}{a^{6}} + \frac {60 \, b^{7} p \log \left (b \sqrt {x}\right )}{a^{6}} + \frac {60 \, {\left (b \sqrt {x} + a\right )}^{5} b^{7} p - 330 \, {\left (b \sqrt {x} + a\right )}^{4} a b^{7} p + 740 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2} b^{7} p - 855 \, {\left (b \sqrt {x} + a\right )}^{2} a^{3} b^{7} p + 522 \, {\left (b \sqrt {x} + a\right )} a^{4} b^{7} p - 137 \, a^{5} b^{7} p + 60 \, a^{5} b^{7} \log \left (c\right )}{{\left (b \sqrt {x} + a\right )}^{6} a^{5} - 6 \, {\left (b \sqrt {x} + a\right )}^{5} a^{6} + 15 \, {\left (b \sqrt {x} + a\right )}^{4} a^{7} - 20 \, {\left (b \sqrt {x} + a\right )}^{3} a^{8} + 15 \, {\left (b \sqrt {x} + a\right )}^{2} a^{9} - 6 \, {\left (b \sqrt {x} + a\right )} a^{10} + a^{11}}}{180 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.56, size = 97, normalized size = 0.75 \begin {gather*} \frac {2\,b^6\,p\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{3\,a^6}-\frac {\frac {b\,p}{5\,a}-\frac {b^2\,p\,\sqrt {x}}{4\,a^2}+\frac {b^5\,p\,x^2}{a^5}-\frac {b^4\,p\,x^{3/2}}{2\,a^4}+\frac {b^3\,p\,x}{3\,a^3}}{3\,x^{5/2}}-\frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________