Optimal. Leaf size=172 \[ -\frac {10 c \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}+\frac {10 c \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \sqrt {b d+2 c d x}}-\frac {20 c}{d \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {687, 693, 694, 329, 298, 203, 206} \[ -\frac {10 c \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}+\frac {10 c \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \sqrt {b d+2 c d x}}-\frac {20 c}{d \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 298
Rule 329
Rule 687
Rule 693
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx &=-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {(5 c) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {(5 c) \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^2}\\ &=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {5 \operatorname {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 \left (b^2-4 a c\right )^2 d^3}\\ &=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {5 \operatorname {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 )}{\left (b^2-4 a c\right )^2 d^3}\\ &=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {(10 c) \operatorname {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}-\frac {(10 c) \operatorname {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}\\ &=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {10 c \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^{3/2}}+\frac {10 c \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^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 55, normalized size = 0.32 \[ -\frac {16 c \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d \left (b^2-4 a c\right )^2 \sqrt {d (b+2 c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.75, size = 1710, normalized size = 9.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.29, size = 650, normalized size = 3.78 \[ \frac {5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} + \frac {5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} + \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} - \frac {4 \, {\left (4 \, b^{2} c d^{2} - 16 \, a c^{2} d^{2} - 5 \, {\left (2 \, c d x + b d\right )}^{2} c\right )}}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} {\left (\sqrt {2 \, c d x + b d} b^{2} d^{2} - 4 \, \sqrt {2 \, c d x + b d} a c d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 404, normalized size = 2.35 \[ \frac {5 \sqrt {2}\, c \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {5 \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {5 \sqrt {2}\, c \ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {4 \left (2 c d x +b d \right )^{\frac {3}{2}} c}{\left (4 a c -b^{2}\right )^{2} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right ) d}-\frac {16 c}{\left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}\, d} \]
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 = 0.74, size = 264, normalized size = 1.53 \[ \frac {\frac {16\,c\,d}{4\,a\,c-b^2}+\frac {20\,c\,{\left (b\,d+2\,c\,d\,x\right )}^2}{d\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {b\,d+2\,c\,d\,x}\,\left (b^2\,d^2-4\,a\,c\,d^2\right )-{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}-\frac {10\,c\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}}-\frac {c\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,10{}\mathrm {i}}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}} \]
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]
________________________________________________________________________________________