Optimal. Leaf size=306 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (B d-A e)}{e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{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)^3 (B d-A e)}{e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
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 {\left (a b+b^2 x\right )^3 (A+B x)}{\sqrt {d+e x}} \, 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 (-B d+A e)}{e^4 \sqrt {d+e x}}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) \sqrt {d+e x}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{5/2}}{e^4}+\frac {b^6 B (d+e x)^{7/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B 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^3 B (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 = 0.10, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} \sqrt {d+e x} \left (-45 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+189 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-105 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+315 (b d-a e)^3 (B d-A e)+35 b^3 B (d+e x)^4\right )}{315 e^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 38.83, size = 374, normalized size = 1.22 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (315 a^3 A e^4+105 a^3 B e^3 (d+e x)-315 a^3 B d e^3+315 a^2 A b e^3 (d+e x)-945 a^2 A b d e^3+945 a^2 b B d^2 e^2-630 a^2 b B d e^2 (d+e x)+189 a^2 b B e^2 (d+e x)^2+945 a A b^2 d^2 e^2-630 a A b^2 d e^2 (d+e x)+189 a A b^2 e^2 (d+e x)^2-945 a b^2 B d^3 e+945 a b^2 B d^2 e (d+e x)-567 a b^2 B d e (d+e x)^2+135 a b^2 B e (d+e x)^3-315 A b^3 d^3 e+315 A b^3 d^2 e (d+e x)-189 A b^3 d e (d+e x)^2+45 A b^3 e (d+e x)^3+315 b^3 B d^4-420 b^3 B d^3 (d+e x)+378 b^3 B d^2 (d+e x)^2-180 b^3 B d (d+e x)^3+35 b^3 B (d+e x)^4\right )}{315 e^4 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 263, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{2} e^{2} - 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 72 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} 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 = 0.23, size = 393, normalized size = 1.28 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{3} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\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 )} B a^{2} b e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\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 a b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 27 \, {\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 a b^{2} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 9 \, {\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 b^{3} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} A 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 = 0.06, size = 317, normalized size = 1.04 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (35 b^{3} B \,x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} 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 [A] time = 0.63, size = 382, normalized size = 1.25 \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 = 2.74, size = 434, normalized size = 1.42 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {-420\,B\,a^3\,d^2\,e^3+630\,A\,a^3\,d\,e^4+1008\,B\,a^2\,b\,d^3\,e^2-1260\,A\,a^2\,b\,d^2\,e^3-864\,B\,a\,b^2\,d^4\,e+1008\,A\,a\,b^2\,d^3\,e^2+256\,B\,b^3\,d^5-288\,A\,b^3\,d^4\,e}{315\,b\,e^5}+\frac {2\,B\,b^2\,x^5}{9}+\frac {x^3\,\left (378\,B\,a^2\,b\,e^5-54\,B\,a\,b^2\,d\,e^4+378\,A\,a\,b^2\,e^5+16\,B\,b^3\,d^2\,e^3-18\,A\,b^3\,d\,e^4\right )}{315\,b\,e^5}+\frac {x\,\left (-210\,B\,a^3\,d\,e^4+630\,A\,a^3\,e^5+504\,B\,a^2\,b\,d^2\,e^3-630\,A\,a^2\,b\,d\,e^4-432\,B\,a\,b^2\,d^3\,e^2+504\,A\,a\,b^2\,d^2\,e^3+128\,B\,b^3\,d^4\,e-144\,A\,b^3\,d^3\,e^2\right )}{315\,b\,e^5}+\frac {x^2\,\left (210\,B\,a^3\,e^5-126\,B\,a^2\,b\,d\,e^4+630\,A\,a^2\,b\,e^5+108\,B\,a\,b^2\,d^2\,e^3-126\,A\,a\,b^2\,d\,e^4-32\,B\,b^3\,d^3\,e^2+36\,A\,b^3\,d^2\,e^3\right )}{315\,b\,e^5}+\frac {2\,b\,x^4\,\left (9\,A\,b\,e+27\,B\,a\,e-B\,b\,d\right )}{63\,e}\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]
________________________________________________________________________________________