Optimal. Leaf size=262 \[ \frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}{e^5 (a+b x)}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^5 (a+b x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^5 (a+b x)}+\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}{e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{\sqrt {d+e x}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{\sqrt {d+e x}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 \sqrt {d+e x}}-\frac {4 b (b d-a e)^3 \sqrt {d+e x}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{5/2}}{e^4}+\frac {b^4 (d+e x)^{7/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 171, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (315 a^4 e^4+420 a^3 b e^3 (e x-2 d)+126 a^2 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+36 a b^3 e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 21.79, size = 241, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (315 a^4 e^4+420 a^3 b e^3 (d+e x)-1260 a^3 b d e^3+1890 a^2 b^2 d^2 e^2+378 a^2 b^2 e^2 (d+e x)^2-1260 a^2 b^2 d e^2 (d+e x)-1260 a b^3 d^3 e+1260 a b^3 d^2 e (d+e x)+180 a b^3 e (d+e x)^3-756 a b^3 d e (d+e x)^2+315 b^4 d^4-420 b^4 d^3 (d+e x)+378 b^4 d^2 (d+e x)^2+35 b^4 (d+e x)^4-180 b^4 d (d+e x)^3\right )}{315 e^4 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 182, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 576 \, a b^{3} d^{3} e + 1008 \, a^{2} b^{2} d^{2} e^{2} - 840 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - 20 \, {\left (2 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{2} e^{2} - 36 \, a b^{3} d e^{3} + 63 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{3} e - 72 \, a b^{3} d^{2} e^{2} + 126 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 244, normalized size = 0.93 \begin {gather*} \frac {2}{315} \, {\left (420 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 36 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} a^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 202, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (35 b^{4} e^{4} x^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 a^{4} e^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.63, size = 382, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 56 \, a b^{2} d^{3} e - 70 \, a^{2} b d^{2} e^{2} + 35 \, a^{3} d e^{3} - {\left (b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (2 \, b^{3} d^{2} e^{2} - 7 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (8 \, b^{3} d^{3} e - 28 \, a b^{2} d^{2} e^{2} + 35 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} a}{35 \, \sqrt {e x + d} e^{4}} + \frac {2 \, {\left (35 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 432 \, a b^{2} d^{4} e + 504 \, a^{2} b d^{3} e^{2} - 210 \, a^{3} d^{2} e^{3} - 5 \, {\left (b^{3} d e^{4} - 27 \, a b^{2} e^{5}\right )} x^{4} + {\left (8 \, b^{3} d^{2} e^{3} - 27 \, a b^{2} d e^{4} + 189 \, a^{2} b e^{5}\right )} x^{3} - {\left (16 \, b^{3} d^{3} e^{2} - 54 \, a b^{2} d^{2} e^{3} + 63 \, a^{2} b d e^{4} - 105 \, a^{3} e^{5}\right )} x^{2} + {\left (64 \, b^{3} d^{4} e - 216 \, a b^{2} d^{3} e^{2} + 252 \, a^{2} b d^{2} e^{3} - 105 \, a^{3} d e^{4}\right )} x\right )} b}{315 \, \sqrt {e x + d} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.50, size = 285, normalized size = 1.09 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^3\,x^5}{9}+\frac {630\,a^4\,d\,e^4-1680\,a^3\,b\,d^2\,e^3+2016\,a^2\,b^2\,d^3\,e^2-1152\,a\,b^3\,d^4\,e+256\,b^4\,d^5}{315\,b\,e^5}+\frac {x\,\left (630\,a^4\,e^5-840\,a^3\,b\,d\,e^4+1008\,a^2\,b^2\,d^2\,e^3-576\,a\,b^3\,d^3\,e^2+128\,b^4\,d^4\,e\right )}{315\,b\,e^5}+\frac {x^2\,\left (840\,a^3\,b\,e^5-252\,a^2\,b^2\,d\,e^4+144\,a\,b^3\,d^2\,e^3-32\,b^4\,d^3\,e^2\right )}{315\,b\,e^5}+\frac {2\,b^2\,x^4\,\left (36\,a\,e-b\,d\right )}{63\,e}+\frac {4\,b\,x^3\,\left (189\,a^2\,e^2-18\,a\,b\,d\,e+4\,b^2\,d^2\right )}{315\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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]
________________________________________________________________________________________