Optimal. Leaf size=256 \[ \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {117 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}-\frac {117 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 256, 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 = {701, 707, 708,
335, 304, 209, 212} \begin {gather*} \frac {117 c^2 \text {ArcTan}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac {117 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac {234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt {b d+2 c d x}}+\frac {234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac {13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 335
Rule 701
Rule 707
Rule 708
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}-\frac {(13 c) \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^3 d^2}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^4 d^4}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {(117 c) \text {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{4 \left (b^2-4 a c\right )^4 d^5}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {(117 c) \text {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{2 \left (b^2-4 a c\right )^4 d^5}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {\left (117 c^2\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 (b^2-4 a c\right )^4 d^3}+\frac {\left (117 c^2\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 (b^2-4 a c\right )^4 d^3}\\ &=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {117 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}-\frac {117 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.64, size = 355, normalized size = 1.39 \begin {gather*} \frac {\left (\frac {1}{10}+\frac {i}{10}\right ) c^2 \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (b+2 c x) \left (-32 b^6+384 a b^4 c-1536 a^2 b^2 c^2+2048 a^3 c^3-416 b^4 (b+2 c x)^2+3328 a b^2 c (b+2 c x)^2-6656 a^2 c^2 (b+2 c x)^2+1053 b^2 (b+2 c x)^4-4212 a c (b+2 c x)^4-585 (b+2 c x)^6\right )}{c^2 \left (b^2-4 a c\right )^4 (a+x (b+c x))^2}-\frac {585 (b+2 c x)^{7/2} \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}+\frac {585 (b+2 c x)^{7/2} \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}-\frac {585 (b+2 c x)^{7/2} \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 )}{\left (b^2-4 a c\right )^{17/4}}\right )}{(d (b+2 c x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.86, size = 382, normalized size = 1.49
method | result | size |
derivativedivides | \(64 d^{5} c^{2} \left (-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}+\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a c \,d^{2}-\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (4 a c \,d^{2}-b^{2} d^{2}+\left (2 c d x +b d \right )^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}\right )\) | \(382\) |
default | \(64 d^{5} c^{2} \left (-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}+\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a c \,d^{2}-\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (4 a c \,d^{2}-b^{2} d^{2}+\left (2 c d x +b d \right )^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}\right )\) | \(382\) |
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. 4347 vs.
\(2 (222) = 444\).
time = 2.33, size = 4347, normalized size = 16.98 \begin {gather*} \text {Too large to display} \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. 958 vs.
\(2 (222) = 444\).
time = 2.46, size = 958, normalized size = 3.74 \begin {gather*} -\frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \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 )}{\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}} - \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \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 )}{\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}} + \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \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 )}{2 \, {\left (\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}\right )}} - \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \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 )}{2 \, {\left (\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}\right )}} - \frac {2 \, {\left (25 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} - 100 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{2} - 21 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2}\right )}}{{\left (b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} + \frac {64 \, {\left (b^{2} c^{2} d^{2} - 4 \, a c^{3} d^{2} + 15 \, {\left (2 \, c d x + b d\right )}^{2} c^{2}\right )}}{5 \, {\left (b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}\right )} {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.00, size = 543, normalized size = 2.12 \begin {gather*} \frac {\frac {234\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^6}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}-\frac {2106\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^4}{5\,\left (-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6\right )}-\frac {64\,c^2\,d^3}{5\,\left (4\,a\,c-b^2\right )}+\frac {832\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^2}{5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4+b^4\,d^4\right )-{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}+\frac {117\,c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}+256\,a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}+96\,a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-256\,a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}-16\,a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}}+\frac {c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}+a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,96{}\mathrm {i}-a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}-a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )\,117{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________