Optimal. Leaf size=161 \[ -\frac {2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e-\left (b^2 \left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac {4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac {4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {698} \[ -\frac {2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e+b^2 \left (-\left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac {4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac {4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx &=\int \left (\frac {\left (b^2 c-a b d+a^2 e\right )^2}{b^4 \sqrt {a+b x}}+\frac {2 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) \sqrt {a+b x}}{b^4}+\frac {\left (-6 a b d e+6 a^2 e^2+b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{3/2}}{b^4}+\frac {2 e (b d-2 a e) (a+b x)^{5/2}}{b^4}+\frac {e^2 (a+b x)^{7/2}}{b^4}\right ) \, dx\\ &=\frac {2 \left (b^2 c-a b d+a^2 e\right )^2 \sqrt {a+b x}}{b^5}+\frac {4 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) (a+b x)^{3/2}}{3 b^5}-\frac {2 \left (6 a b d e-6 a^2 e^2-b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{5/2}}{5 b^5}+\frac {4 e (b d-2 a e) (a+b x)^{7/2}}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 155, normalized size = 0.96 \[ \frac {2 \sqrt {a+b x} \left (128 a^4 e^2-32 a^3 b e (9 d+2 e x)+24 a^2 b^2 \left (2 e \left (7 c+e x^2\right )+7 d^2+6 d e x\right )-4 a b^3 \left (21 c (5 d+2 e x)+x \left (21 d^2+27 d e x+10 e^2 x^2\right )\right )+b^4 \left (315 c^2+42 c x (5 d+3 e x)+x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )\right )}{315 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 192, normalized size = 1.19 \[ \frac {2 \, {\left (35 \, b^{4} e^{2} x^{4} + 315 \, b^{4} c^{2} - 420 \, a b^{3} c d + 168 \, a^{2} b^{2} d^{2} + 128 \, a^{4} e^{2} + 10 \, {\left (9 \, b^{4} d e - 4 \, a b^{3} e^{2}\right )} x^{3} + 3 \, {\left (21 \, b^{4} d^{2} + 16 \, a^{2} b^{2} e^{2} + 6 \, {\left (7 \, b^{4} c - 6 \, a b^{3} d\right )} e\right )} x^{2} + 48 \, {\left (7 \, a^{2} b^{2} c - 6 \, a^{3} b d\right )} e + 2 \, {\left (105 \, b^{4} c d - 42 \, a b^{3} d^{2} - 32 \, a^{3} b e^{2} - 12 \, {\left (7 \, a b^{3} c - 6 \, a^{2} b^{2} d\right )} e\right )} x\right )} \sqrt {b x + a}}{315 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 237, normalized size = 1.47 \[ \frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + \frac {210 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} c d}{b} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {42 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} c e}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 194, normalized size = 1.20 \[ \frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 c^{2} b^{4}\right )}{315 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.90, size = 237, normalized size = 1.47 \[ \frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + 42 \, c {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.76, size = 149, normalized size = 0.93 \[ \frac {2\,e^2\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (12\,a^2\,e^2-12\,a\,b\,d\,e+2\,b^2\,d^2+4\,c\,b^2\,e\right )}{5\,b^5}+\frac {2\,\sqrt {a+b\,x}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^2}{b^5}-\frac {\left (8\,a\,e^2-4\,b\,d\,e\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {4\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{3/2}\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 85.15, size = 644, normalized size = 4.00 \[ \begin {cases} \frac {- \frac {2 a c^{2}}{\sqrt {a + b x}} - \frac {4 a c d \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {4 a c e \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} - \frac {2 a d^{2} \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} - \frac {4 a d e \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{3}} - \frac {2 a e^{2} \left (\frac {a^{4}}{\sqrt {a + b x}} + 4 a^{3} \sqrt {a + b x} - 2 a^{2} \left (a + b x\right )^{\frac {3}{2}} + \frac {4 a \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {\left (a + b x\right )^{\frac {7}{2}}}{7}\right )}{b^{4}} - 2 c^{2} \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right ) - \frac {4 c d \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b} - \frac {4 c e \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}} - \frac {2 d^{2} \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}} - \frac {4 d e \left (\frac {a^{4}}{\sqrt {a + b x}} + 4 a^{3} \sqrt {a + b x} - 2 a^{2} \left (a + b x\right )^{\frac {3}{2}} + \frac {4 a \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {\left (a + b x\right )^{\frac {7}{2}}}{7}\right )}{b^{3}} - \frac {2 e^{2} \left (- \frac {a^{5}}{\sqrt {a + b x}} - 5 a^{4} \sqrt {a + b x} + \frac {10 a^{3} \left (a + b x\right )^{\frac {3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac {5}{2}} + \frac {5 a \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {\left (a + b x\right )^{\frac {9}{2}}}{9}\right )}{b^{4}}}{b} & \text {for}\: b \neq 0 \\\frac {c^{2} x + c d x^{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{5}}{5} + \frac {x^{3} \left (2 c e + d^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________