Optimal. Leaf size=223 \[ \frac {45 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}-\frac {45 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}+\frac {90 c^2}{d \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}+\frac {9 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \sqrt {b d+2 c d x}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \sqrt {b d+2 c d x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} \frac {45 c^2 \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 )^{13/4}}-\frac {45 c^2 \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 )^{13/4}}+\frac {90 c^2}{d \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}+\frac {9 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \sqrt {b d+2 c d x}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \sqrt {b d+2 c d x}} \end {gather*}
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 )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}-\frac {(9 c) \int \frac {1}{(b d+2 c d x)^{3/2} \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 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {\left (45 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 )^2}\\ &=\frac {90 c^2}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {\left (45 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 )^3 d^2}\\ &=\frac {90 c^2}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {(45 c) \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 )}{4 \left (b^2-4 a c\right )^3 d^3}\\ &=\frac {90 c^2}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {(45 c) \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 )}{2 \left (b^2-4 a c\right )^3 d^3}\\ &=\frac {90 c^2}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {\left (45 c^2\right ) \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 )^3 d}+\frac {\left (45 c^2\right ) \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 )^3 d}\\ &=\frac {90 c^2}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2}+\frac {9 c}{2 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {45 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 )^{13/4} d^{3/2}}-\frac {45 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 )^{13/4} d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 57, normalized size = 0.26 \begin {gather*} \frac {64 c^2 \, _2F_1\left (-\frac {1}{4},3;\frac {3}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d \left (b^2-4 a c\right )^3 \sqrt {d (b+2 c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 1.61, size = 331, normalized size = 1.48 \begin {gather*} \frac {\left (128 a^2 c^2+17 a b^2 c+324 a b c^2 x+324 a c^3 x^2-b^4+9 b^3 c x+189 b^2 c^2 x^2+360 b c^3 x^3+180 c^4 x^4\right ) \sqrt {b d+2 c d x}}{2 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \left (a+b x+c x^2\right )^2}-\frac {\left (\frac {45}{2}+\frac {45 i}{2}\right ) c^2 \tan ^{-1}\left (\frac {-\frac {(1+i) c \sqrt {d} x}{\sqrt [4]{b^2-4 a c}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {d}}{\sqrt [4]{b^2-4 a c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {d} \sqrt [4]{b^2-4 a c}}{\sqrt {b d+2 c d x}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}-\frac {\left (\frac {45}{2}+\frac {45 i}{2}\right ) c^2 \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b d+2 c d x}}{\sqrt {d} \left (\sqrt {b^2-4 a c}+i b+2 i c x\right )}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 2890, normalized size = 12.96
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.35, size = 813, normalized size = 3.65 \begin {gather*} -\frac {45 \, {\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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}} - \frac {45 \, {\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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}} + \frac {45 \, {\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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}\right )}} - \frac {45 \, {\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^{8} d^{3} - 16 \, \sqrt {2} a b^{6} c d^{3} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{3} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{3} + 256 \, \sqrt {2} a^{4} c^{4} d^{3}\right )}} + \frac {64 \, c^{2}}{{\left (b^{6} d - 12 \, a b^{4} c d + 48 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3} d\right )} \sqrt {2 \, c d x + b d}} - \frac {2 \, {\left (17 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} - 68 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{2} - 13 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2}\right )}}{{\left (b^{6} d - 12 \, a b^{4} c d + 48 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3} d\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 534, normalized size = 2.39 \begin {gather*} -\frac {136 \left (2 c d x +b d \right )^{\frac {3}{2}} a \,c^{3} d}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {34 \left (2 c d x +b d \right )^{\frac {3}{2}} b^{2} c^{2} d}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {45 \sqrt {2}\, c^{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 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {45 \sqrt {2}\, c^{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 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {45 \sqrt {2}\, c^{2} \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 )}{4 \left (4 a c -b^{2}\right )^{3} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d}-\frac {26 \left (2 c d x +b d \right )^{\frac {7}{2}} c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2} d}-\frac {64 c^{2}}{\left (4 a c -b^{2}\right )^{3} \sqrt {2 c d x +b d}\, d} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 0.88, size = 421, normalized size = 1.89 \begin {gather*} \frac {45\,c^2\,\mathrm {atan}\left (\frac {b^6\,\sqrt {b\,d+2\,c\,d\,x}-64\,a^3\,c^3\,\sqrt {b\,d+2\,c\,d\,x}+48\,a^2\,b^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-12\,a\,b^4\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{13/4}}\right )}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{13/4}}-\frac {\frac {64\,c^2\,d^3}{4\,a\,c-b^2}-\frac {90\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^4}{-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6}+\frac {162\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^2}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{\sqrt {b\,d+2\,c\,d\,x}\,\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 )}^{5/2}\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}+\frac {c^2\,\mathrm {atan}\left (\frac {b^6\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a^3\,c^3\,\sqrt {b\,d+2\,c\,d\,x}\,64{}\mathrm {i}+a^2\,b^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,48{}\mathrm {i}-a\,b^4\,c\,\sqrt {b\,d+2\,c\,d\,x}\,12{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{13/4}}\right )\,45{}\mathrm {i}}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{13/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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]
________________________________________________________________________________________