Optimal. Leaf size=210 \[ 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {700, 706, 708,
335, 218, 212, 209} \begin {gather*} -26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \text {ArcTan}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+52 c d^7 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 706
Rule 708
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx\\ &=\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\\ &=\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^3 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {1}{2} \left (13 \left (b^2-4 a c\right )^3 d^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 \left (b^2-4 a c\right )^3 d^7\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.13, size = 351, normalized size = 1.67 \begin {gather*} \left (\frac {1}{45}+\frac {i}{45}\right ) c (d (b+2 c x))^{15/2} \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-585 b^6+7020 a b^4 c-28080 a^2 b^2 c^2+37440 a^3 c^3+468 b^4 (b+2 c x)^2-3744 a b^2 c (b+2 c x)^2+7488 a^2 c^2 (b+2 c x)^2+52 b^2 (b+2 c x)^4-208 a c (b+2 c x)^4+20 (b+2 c x)^6\right )}{c (b+2 c x)^7 (a+x (b+c x))}-\frac {585 i \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{15/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(457\) vs.
\(2(178)=356\).
time = 0.85, size = 458, normalized size = 2.18
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (48 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-24 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+3 b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-d^{6} \left (\frac {\left (-4 a^{3} c^{3}+3 a^{2} b^{2} c^{2}-\frac {3}{4} a \,b^{4} c +\frac {1}{16} b^{6}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {13 \left (16 a^{3} c^{3}-12 a^{2} b^{2} c^{2}+3 a \,b^{4} c -\frac {1}{4} b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(458\) |
default | \(16 c \,d^{3} \left (48 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-24 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+3 b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-d^{6} \left (\frac {\left (-4 a^{3} c^{3}+3 a^{2} b^{2} c^{2}-\frac {3}{4} a \,b^{4} c +\frac {1}{16} b^{6}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {13 \left (16 a^{3} c^{3}-12 a^{2} b^{2} c^{2}+3 a \,b^{4} c -\frac {1}{4} b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(458\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1445 vs.
\(2 (178) = 356\).
time = 3.23, size = 1445, normalized size = 6.88 \begin {gather*} -\frac {2340 \, \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \arctan \left (\frac {\left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {3}{4}} {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} \sqrt {2 \, c d x + b d} d^{7} - \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {3}{4}} \sqrt {2 \, {\left (b^{8} c^{3} - 16 \, a b^{6} c^{4} + 96 \, a^{2} b^{4} c^{5} - 256 \, a^{3} b^{2} c^{6} + 256 \, a^{4} c^{7}\right )} d^{15} x + {\left (b^{9} c^{2} - 16 \, a b^{7} c^{3} + 96 \, a^{2} b^{5} c^{4} - 256 \, a^{3} b^{3} c^{5} + 256 \, a^{4} b c^{6}\right )} d^{15} + \sqrt {{\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}}}}{{\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}}\right ) + 585 \, \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (13 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} \sqrt {2 \, c d x + b d} d^{7} + 13 \, \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {1}{4}}\right ) - 585 \, \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (13 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} \sqrt {2 \, c d x + b d} d^{7} - 13 \, \left ({\left (b^{18} c^{4} - 36 \, a b^{16} c^{5} + 576 \, a^{2} b^{14} c^{6} - 5376 \, a^{3} b^{12} c^{7} + 32256 \, a^{4} b^{10} c^{8} - 129024 \, a^{5} b^{8} c^{9} + 344064 \, a^{6} b^{6} c^{10} - 589824 \, a^{7} b^{4} c^{11} + 589824 \, a^{8} b^{2} c^{12} - 262144 \, a^{9} c^{13}\right )} d^{30}\right )^{\frac {1}{4}}\right ) - {\left (1280 \, c^{6} d^{7} x^{6} + 3840 \, b c^{5} d^{7} x^{5} + 256 \, {\left (22 \, b^{2} c^{4} - 13 \, a c^{5}\right )} d^{7} x^{4} + 256 \, {\left (19 \, b^{3} c^{3} - 26 \, a b c^{4}\right )} d^{7} x^{3} + 96 \, {\left (45 \, b^{4} c^{2} - 208 \, a b^{2} c^{3} + 312 \, a^{2} c^{4}\right )} d^{7} x^{2} + 32 \, {\left (79 \, b^{5} c - 520 \, a b^{3} c^{2} + 936 \, a^{2} b c^{3}\right )} d^{7} x - {\left (45 \, b^{6} - 3068 \, a b^{4} c + 20592 \, a^{2} b^{2} c^{2} - 37440 \, a^{3} c^{3}\right )} d^{7}\right )} \sqrt {2 \, c d x + b d}}{45 \, {\left (c x^{2} + b x + a\right )}} \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. 876 vs.
\(2 (178) = 356\).
time = 2.60, size = 876, normalized size = 4.17 \begin {gather*} 48 \, \sqrt {2 \, c d x + b d} b^{4} c d^{7} - 384 \, \sqrt {2 \, c d x + b d} a b^{2} c^{2} d^{7} + 768 \, \sqrt {2 \, c d x + b d} a^{2} c^{3} d^{7} + \frac {32}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c d^{5} - \frac {128}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{2} d^{5} + \frac {16}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c d^{3} - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {4 \, {\left (\sqrt {2 \, c d x + b d} b^{6} c d^{9} - 12 \, \sqrt {2 \, c d x + b d} a b^{4} c^{2} d^{9} + 48 \, \sqrt {2 \, c d x + b d} a^{2} b^{2} c^{3} d^{9} - 64 \, \sqrt {2 \, c d x + b d} a^{3} c^{4} d^{9}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.69, size = 1060, normalized size = 5.05 \begin {gather*} \frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (-256\,a^3\,c^4\,d^9+192\,a^2\,b^2\,c^3\,d^9-48\,a\,b^4\,c^2\,d^9+4\,b^6\,c\,d^9\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}+48\,c\,d^7\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-26\,c\,d^{15/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}-\frac {32\,c\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}-c\,d^{15/2}\,\mathrm {atan}\left (\frac {c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}+c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}}{13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,26{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________