Optimal. Leaf size=302 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) \sqrt {d+e x}}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)} \]
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Rubi [A] time = 0.14, antiderivative size = 302, 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)^{5/2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-a B e-A b e+2 b B d)}{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)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) \sqrt {d+e x}}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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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}}{(d+e x)^{3/2}} \, 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)}{(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 (-B d+A e)}{e^4 (d+e x)^{3/2}}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 \sqrt {d+e x}}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) \sqrt {d+e x}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{3/2}}{e^4}+\frac {b^6 B (d+e x)^{5/2}}{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 {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 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^3 B (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 240, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (35 a^3 e^3 (-A e+2 B d+B e x)+35 a^2 b e^2 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+7 a b^2 e \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )\right )}{35 e^5 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 23.91, size = 374, normalized size = 1.24 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-35 a^3 A e^4+35 a^3 B e^3 (d+e x)+35 a^3 B d e^3+105 a^2 A b e^3 (d+e x)+105 a^2 A b d e^3-105 a^2 b B d^2 e^2-210 a^2 b B d e^2 (d+e x)+35 a^2 b B e^2 (d+e x)^2-105 a A b^2 d^2 e^2-210 a A b^2 d e^2 (d+e x)+35 a A b^2 e^2 (d+e x)^2+105 a b^2 B d^3 e+315 a b^2 B d^2 e (d+e x)-105 a b^2 B d e (d+e x)^2+21 a b^2 B e (d+e x)^3+35 A b^3 d^3 e+105 A b^3 d^2 e (d+e x)-35 A b^3 d e (d+e x)^2+7 A b^3 e (d+e x)^3-35 b^3 B d^4-140 b^3 B d^3 (d+e x)+70 b^3 B d^2 (d+e x)^2-28 b^3 B d (d+e x)^3+5 b^3 B (d+e x)^4\right )}{35 e^4 \sqrt {d+e x} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 272, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 35 \, A a^{3} e^{4} + 112 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 280 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - {\left (8 \, B b^{3} d e^{3} - 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + {\left (16 \, B b^{3} d^{2} e^{2} - 14 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 35 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 56 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 140 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 525, normalized size = 1.74 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{3} e^{30} \mathrm {sgn}\left (b x + a\right ) - 28 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{30} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{30} \mathrm {sgn}\left (b x + a\right ) - 140 \, \sqrt {x e + d} B b^{3} d^{3} e^{30} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{31} \mathrm {sgn}\left (b x + a\right ) + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{31} \mathrm {sgn}\left (b x + a\right ) - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{31} \mathrm {sgn}\left (b x + a\right ) - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{31} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} B a b^{2} d^{2} e^{31} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} A b^{3} d^{2} e^{31} \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{32} \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{32} \mathrm {sgn}\left (b x + a\right ) - 210 \, \sqrt {x e + d} B a^{2} b d e^{32} \mathrm {sgn}\left (b x + a\right ) - 210 \, \sqrt {x e + d} A a b^{2} d e^{32} \mathrm {sgn}\left (b x + a\right ) + 35 \, \sqrt {x e + d} B a^{3} e^{33} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} A a^{2} b e^{33} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-35\right )} - \frac {2 \, {\left (B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) - A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 317, normalized size = 1.05 \begin {gather*} -\frac {2 \left (-5 b^{3} B \,x^{4} e^{4}-7 A \,b^{3} e^{4} x^{3}-21 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-35 A a \,b^{2} e^{4} x^{2}+14 A \,b^{3} d \,e^{3} x^{2}-35 B \,a^{2} b \,e^{4} x^{2}+42 B a \,b^{2} d \,e^{3} x^{2}-16 B \,b^{3} d^{2} e^{2} x^{2}-105 A \,a^{2} b \,e^{4} x +140 A a \,b^{2} d \,e^{3} x -56 A \,b^{3} d^{2} e^{2} x -35 B \,a^{3} e^{4} x +140 B \,a^{2} b d \,e^{3} x -168 B a \,b^{2} d^{2} e^{2} x +64 B \,b^{3} d^{3} e x +35 A \,a^{3} e^{4}-210 A \,a^{2} b d \,e^{3}+280 A a \,b^{2} d^{2} e^{2}-112 A \,b^{3} d^{3} e -70 B \,a^{3} d \,e^{3}+280 B \,a^{2} b \,d^{2} e^{2}-336 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}}}{35 \sqrt {e x +d}\, \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 282, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - {\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} A}{5 \, \sqrt {e x + d} e^{4}} + \frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 336 \, a b^{2} d^{3} e - 280 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} - {\left (8 \, b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (16 \, b^{3} d^{2} e^{2} - 42 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (64 \, b^{3} d^{3} e - 168 \, a b^{2} d^{2} e^{2} + 140 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} B}{35 \, \sqrt {e x + d} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 327, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x\,\left (70\,B\,a^3\,e^4-280\,B\,a^2\,b\,d\,e^3+210\,A\,a^2\,b\,e^4+336\,B\,a\,b^2\,d^2\,e^2-280\,A\,a\,b^2\,d\,e^3-128\,B\,b^3\,d^3\,e+112\,A\,b^3\,d^2\,e^2\right )}{35\,b\,e^5}-\frac {-140\,B\,a^3\,d\,e^3+70\,A\,a^3\,e^4+560\,B\,a^2\,b\,d^2\,e^2-420\,A\,a^2\,b\,d\,e^3-672\,B\,a\,b^2\,d^3\,e+560\,A\,a\,b^2\,d^2\,e^2+256\,B\,b^3\,d^4-224\,A\,b^3\,d^3\,e}{35\,b\,e^5}+\frac {x^2\,\left (70\,B\,a^2\,b\,e^4-84\,B\,a\,b^2\,d\,e^3+70\,A\,a\,b^2\,e^4+32\,B\,b^3\,d^2\,e^2-28\,A\,b^3\,d\,e^3\right )}{35\,b\,e^5}+\frac {2\,b\,x^3\,\left (7\,A\,b\,e+21\,B\,a\,e-8\,B\,b\,d\right )}{35\,e^2}+\frac {2\,B\,b^2\,x^4}{7\,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.
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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.
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