Optimal. Leaf size=743 \[ -\frac {32 b^2 c^2-189 a b c d+117 a^2 d^2}{80 a c^3 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{17/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.83, antiderivative size = 743, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483,
593, 597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{17/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac {117 a^3 d^3-189 a^2 b c d^2+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt {x} (b c-a d)^2}-\frac {d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 303
Rule 477
Rule 483
Rule 593
Rule 597
Rule 598
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-13 a d-13 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-189 a b c d+117 a^2 d^2-9 b d (21 b c-13 a d) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {5 \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right )+5 b d \left (32 b^2 c^2-189 a b c d+117 a^2 d^2\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{80 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (5 \left (32 b^4 c^4+32 a b^3 c^3 d+32 a^2 b^2 c^2 d^2-189 a^3 b c d^3+117 a^4 d^4\right )+5 b d \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {160 b^5 c^4 x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {5 a^2 d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\left (2 b^5\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}-\frac {\left (d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^4 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{9/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}+\frac {b^{9/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}+\frac {\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 (b c-a d)^3}-\frac {\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^4 (b c-a d)^3}-\frac {\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^4 (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{17/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.18, size = 462, normalized size = 0.62 \begin {gather*} \frac {1}{320} \left (\frac {4 \left (160 b^3 c^3 x^2 \left (c+d x^2\right )^2-32 a b^2 c^2 \left (c-5 d x^2\right ) \left (c+d x^2\right )^2+a^2 b c d \left (64 c^3-672 c^2 d x^2-1701 c d^2 x^4-945 d^3 x^6\right )+a^3 d^2 \left (-32 c^3+416 c^2 d x^2+1053 c d^2 x^4+585 d^3 x^6\right )\right )}{a^2 c^4 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac {160 \sqrt {2} b^{17/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (-b c+a d)^3}+\frac {5 \sqrt {2} d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{17/4} (b c-a d)^3}+\frac {160 \sqrt {2} b^{17/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (-b c+a d)^3}+\frac {5 \sqrt {2} d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{17/4} (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.23, size = 368, normalized size = 0.50 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.61, size = 756, normalized size = 1.02 \begin {gather*} \frac {b^{5} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (221 \, b^{2} c^{2} d^{3} - 306 \, a b c d^{4} + 117 \, a^{2} d^{5}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )}} - \frac {32 \, a b^{2} c^{5} - 64 \, a^{2} b c^{4} d + 32 \, a^{3} c^{3} d^{2} - 5 \, {\left (32 \, b^{3} c^{3} d^{2} + 32 \, a b^{2} c^{2} d^{3} - 189 \, a^{2} b c d^{4} + 117 \, a^{3} d^{5}\right )} x^{6} - {\left (320 \, b^{3} c^{4} d + 288 \, a b^{2} c^{3} d^{2} - 1701 \, a^{2} b c^{2} d^{3} + 1053 \, a^{3} c d^{4}\right )} x^{4} - 32 \, {\left (5 \, b^{3} c^{5} + 3 \, a b^{2} c^{4} d - 21 \, a^{2} b c^{3} d^{2} + 13 \, a^{3} c^{2} d^{3}\right )} x^{2}}{80 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{\frac {13}{2}} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{\frac {9}{2}} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
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 [A]
time = 2.92, size = 1000, normalized size = 1.35 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} + \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {29 \, b c d^{4} x^{\frac {7}{2}} - 21 \, a d^{5} x^{\frac {7}{2}} + 33 \, b c^{2} d^{3} x^{\frac {3}{2}} - 25 \, a c d^{4} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {2 \, {\left (5 \, b c x^{2} + 15 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 7.03, size = 2500, normalized size = 3.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________