Optimal. Leaf size=219 \[ \frac {2 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {2 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^4 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^4 \, 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 (a+b x)^6}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^7}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^8}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^9}{b^4}+\frac {e^4 (a+b x)^{10}}{b^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {e (b d-a e)^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {2 e^2 (b d-a e)^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^5}+\frac {2 e^3 (b d-a e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {e^4 (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 371, normalized size = 1.69 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (462 a^6 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+462 a^5 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+330 a^4 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+165 a^3 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+55 a^2 b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+11 a b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+b^6 x^6 \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )\right )}{2310 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 3.44, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 418, normalized size = 1.91 \begin {gather*} \frac {1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac {1}{5} \, {\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} + {\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 666, normalized size = 3.04 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, b^{6} d x^{10} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{6} d^{2} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{6} d^{3} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a b^{5} x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a b^{5} d x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a b^{5} d^{2} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{7} \, a b^{5} d^{3} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{2} b^{4} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{4} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {90}{7} \, a^{2} b^{4} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {80}{7} \, a^{3} b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 16 \, a^{3} b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a^{4} b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{4} b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{4} b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{5} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{5} \, a^{5} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{5} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{5} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{6} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{4} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 489, normalized size = 2.23 \begin {gather*} \frac {\left (210 e^{4} b^{6} x^{10}+1386 x^{9} e^{4} a \,b^{5}+924 x^{9} d \,e^{3} b^{6}+3850 x^{8} e^{4} a^{2} b^{4}+6160 x^{8} d \,e^{3} a \,b^{5}+1540 x^{8} d^{2} e^{2} b^{6}+5775 x^{7} e^{4} a^{3} b^{3}+17325 x^{7} d \,e^{3} a^{2} b^{4}+10395 x^{7} d^{2} e^{2} a \,b^{5}+1155 x^{7} d^{3} e \,b^{6}+4950 x^{6} e^{4} a^{4} b^{2}+26400 x^{6} d \,e^{3} a^{3} b^{3}+29700 x^{6} d^{2} e^{2} a^{2} b^{4}+7920 x^{6} d^{3} e a \,b^{5}+330 x^{6} d^{4} b^{6}+2310 a^{5} b \,e^{4} x^{5}+23100 a^{4} b^{2} d \,e^{3} x^{5}+46200 a^{3} b^{3} d^{2} e^{2} x^{5}+23100 a^{2} b^{4} d^{3} e \,x^{5}+2310 a \,b^{5} d^{4} x^{5}+462 x^{4} e^{4} a^{6}+11088 x^{4} d \,e^{3} a^{5} b +41580 x^{4} d^{2} e^{2} a^{4} b^{2}+36960 x^{4} d^{3} e \,a^{3} b^{3}+6930 x^{4} d^{4} a^{2} b^{4}+2310 a^{6} d \,e^{3} x^{3}+20790 a^{5} b \,d^{2} e^{2} x^{3}+34650 a^{4} b^{2} d^{3} e \,x^{3}+11550 a^{3} b^{3} d^{4} x^{3}+4620 a^{6} d^{2} e^{2} x^{2}+18480 a^{5} b \,d^{3} e \,x^{2}+11550 a^{4} b^{2} d^{4} x^{2}+4620 a^{6} d^{3} e x +6930 a^{5} b \,d^{4} x +2310 d^{4} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{2310 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.53, size = 998, normalized size = 4.56
result too large to display
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 (a+b\,x\right )\,{\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,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 (a + b x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________