Optimal. Leaf size=175 \[ -\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}+\frac {a x^8 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 98, 147, 63, 208} \[ -\frac {\sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}-\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}+\frac {a x^8 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 98
Rule 147
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {x \left (2 a c+\frac {1}{2} (-2 b c+5 a d) x\right )}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{4 b (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac {\left (a^2 (6 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{8 b^3 (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac {\left (a^2 (6 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{4 b^3 d (b c-a d)}\\ &=\frac {a x^8 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt {c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}-\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 175, normalized size = 1.00 \[ \frac {a^2 (5 a d-6 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4} \left (-15 a^3 d^2+2 a^2 b d \left (4 c-5 d x^4\right )+2 a b^2 \left (2 c^2+3 c d x^4+d^2 x^8\right )+2 b^3 c x^4 \left (2 c-d x^4\right )\right )}{12 b^3 d^2 \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.58, size = 622, normalized size = 3.55 \[ \left [\frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}, \frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) + {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 180, normalized size = 1.03 \[ \frac {\sqrt {d x^{4} + c} a^{3} d}{4 \, {\left (b^{4} c - a b^{3} d\right )} {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac {{\left (6 \, a^{2} b c - 5 \, a^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{4} d^{4} - 3 \, \sqrt {d x^{4} + c} b^{4} c d^{4} - 6 \, \sqrt {d x^{4} + c} a b^{3} d^{5}}{6 \, b^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.25, size = 923, normalized size = 5.27 \[ \frac {a^{3} d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, b^{4}}+\frac {a^{3} d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, b^{4}}+\frac {\sqrt {d \,x^{4}+c}\, x^{4}}{6 b^{2} d}-\frac {3 a^{2} \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{4}}-\frac {3 a^{2} \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{4}}-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2}}{8 \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) b^{4}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a^{2}}{8 \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) b^{4}}-\frac {\sqrt {d \,x^{4}+c}\, a}{b^{3} d}-\frac {\sqrt {d \,x^{4}+c}\, c}{3 b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.19, size = 186, normalized size = 1.06 \[ \frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b^2\,d^2}-\left (\frac {3\,c}{2\,b^2\,d^2}+\frac {a\,d-b\,c}{b^3\,d^2}\right )\,\sqrt {d\,x^4+c}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\left (5\,a\,d-6\,b\,c\right )}{\sqrt {a\,d-b\,c}\,\left (5\,a^3\,d-6\,a^2\,b\,c\right )}\right )\,\left (5\,a\,d-6\,b\,c\right )}{4\,b^{7/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a^3\,d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b^4\,\left (d\,x^4+c\right )-2\,b^4\,c+2\,a\,b^3\,d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________