Optimal. Leaf size=208 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^4 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^4 (a+b x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {646, 43} \begin {gather*} \frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^4 (a+b x)}-\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{7/2}}{e^3}+\frac {b^6 (d+e x)^{9/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac {6 b (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}-\frac {2 b^2 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^4 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 120, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (231 a^3 e^3+99 a^2 b e^2 (5 e x-2 d)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 33.85, size = 160, normalized size = 0.77 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (231 a^3 e^3+495 a^2 b e^2 (d+e x)-693 a^2 b d e^2+693 a b^2 d^2 e+385 a b^2 e (d+e x)^2-990 a b^2 d e (d+e x)-231 b^3 d^3+495 b^3 d^2 (d+e x)+105 b^3 (d+e x)^3-385 b^3 d (d+e x)^2\right )}{1155 e^3 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 216, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 670, normalized size = 3.22 \begin {gather*} \frac {2}{3465} \, {\left (3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d^{2} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 693 \, {\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 b^{2} d^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 99 \, {\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 )} b^{3} d^{2} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 1386 \, {\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 d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 594 \, {\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^{2} d e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 22 \, {\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^{3} d e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 3465 \, \sqrt {x e + d} a^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d \mathrm {sgn}\left (b x + a\right ) + 297 \, {\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^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 33 \, {\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 )} a b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 231 \, {\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^{3} \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 = 0.05, size = 132, normalized size = 0.63 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 b^{3} e^{3} x^{3}+385 a \,b^{2} e^{3} x^{2}-70 b^{3} d \,e^{2} x^{2}+495 a^{2} b \,e^{3} x -220 a \,b^{2} d \,e^{2} x +40 b^{3} d^{2} e x +231 a^{3} e^{3}-198 a^{2} b d \,e^{2}+88 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1155 \left (b x +a \right )^{3} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.19, size = 216, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________