Optimal. Leaf size=159 \[ \frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {707, 708, 335,
304, 209, 212} \begin {gather*} \frac {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 )^{9/4}}-\frac {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 )^{9/4}}+\frac {4}{d^3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}+\frac {4}{5 d \left (b^2-4 a c\right ) (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 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 )} \, dx &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^4}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\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 )}{2 c \left (b^2-4 a c\right )^2 d^5}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\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 )}{c \left (b^2-4 a c\right )^2 d^5}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {2 \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 )^2 d^3}+\frac {2 \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 )^2 d^3}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {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 )^{9/4} d^{7/2}}-\frac {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 )^{9/4} d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.33, size = 219, normalized size = 1.38 \begin {gather*} \frac {\left (\frac {1}{5}+\frac {i}{5}\right ) \left ((2-2 i) \sqrt [4]{b^2-4 a c} (b+2 c x) \left (b^2-4 a c+5 (b+2 c x)^2\right )-5 (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 )+5 (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 )-5 (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 )\right )}{\left (b^2-4 a c\right )^{9/4} (d (b+2 c x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(300\) vs.
\(2(133)=266\).
time = 0.71, size = 301, normalized size = 1.89
method | result | size |
derivativedivides | \(4 d \left (-\frac {1}{5 d^{2} \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {1}{d^{4} \left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{4} \left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(301\) |
default | \(4 d \left (-\frac {1}{5 d^{2} \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {1}{d^{4} \left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{4} \left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(301\) |
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. 1693 vs.
\(2 (133) = 266\).
time = 4.56, size = 1693, normalized size = 10.65 \begin {gather*} -\frac {20 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{8} \sqrt {\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}} + 2 \, c d x + b d} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {1}{4}} - {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{3} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {1}{4}}\right ) + 5 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {1}{4}} \log \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{11} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {3}{4}} + \sqrt {2 \, c d x + b d}\right ) - 5 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {1}{4}} \log \left (-{\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{11} \left (\frac {1}{{\left (b^{18} - 36 \, a b^{16} c + 576 \, a^{2} b^{14} c^{2} - 5376 \, a^{3} b^{12} c^{3} + 32256 \, a^{4} b^{10} c^{4} - 129024 \, a^{5} b^{8} c^{5} + 344064 \, a^{6} b^{6} c^{6} - 589824 \, a^{7} b^{4} c^{7} + 589824 \, a^{8} b^{2} c^{8} - 262144 \, a^{9} c^{9}\right )} d^{14}}\right )^{\frac {3}{4}} + \sqrt {2 \, c d x + b d}\right ) - 8 \, {\left (10 \, c^{2} x^{2} + 10 \, b c x + 3 \, b^{2} - 2 \, a c\right )} \sqrt {2 \, c d x + b d}}{5 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\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. 605 vs.
\(2 (133) = 266\).
time = 3.69, size = 605, normalized size = 3.81 \begin {gather*} -\frac {\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} - \frac {\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} + \frac {{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{\sqrt {2} b^{6} d^{5} - 12 \, \sqrt {2} a b^{4} c d^{5} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt {2} a^{3} c^{3} d^{5}} - \frac {{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{\sqrt {2} b^{6} d^{5} - 12 \, \sqrt {2} a b^{4} c d^{5} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt {2} a^{3} c^{3} d^{5}} + \frac {4 \, {\left (b^{2} d^{2} - 4 \, a c d^{2} + 5 \, {\left (2 \, c d x + b d\right )}^{2}\right )}}{5 \, {\left (b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} 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 = 0.26, size = 230, normalized size = 1.45 \begin {gather*} \frac {\frac {4}{5\,\left (b^2\,d-4\,a\,c\,d\right )}+\frac {4\,{\left (b\,d+2\,c\,d\,x\right )}^2}{d\,{\left (b^2\,d-4\,a\,c\,d\right )}^2}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}+\frac {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 )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}}+\frac {\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,2{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________