Optimal. Leaf size=407 \[ -\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.38, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \begin {gather*} -\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right ) (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^2 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{8 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 111, normalized size = 0.27 \begin {gather*} \frac {(a+b x) (d+e x)^{9/2} \left (\frac {e (a+b x)^2 (-9 a B e+5 A b e+4 b B d) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+9 a B-9 A b\right )}{18 b \left ((a+b x)^2\right )^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 64.93, size = 509, normalized size = 1.25 \begin {gather*} \frac {(-a e-b e x) \left (\frac {7 (b d-a e)^2 \left (-9 a B e^2+5 A b e^2+4 b B d e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{11/2} \sqrt {a e-b d}}-\frac {e \sqrt {d+e x} \left (945 a^4 B e^4-525 a^3 A b e^4+1575 a^3 b B e^3 (d+e x)-3255 a^3 b B d e^3-875 a^2 A b^2 e^3 (d+e x)+1575 a^2 A b^2 d e^3+4095 a^2 b^2 B d^2 e^2+504 a^2 b^2 B e^2 (d+e x)^2-3850 a^2 b^2 B d e^2 (d+e x)-1575 a A b^3 d^2 e^2-280 a A b^3 e^2 (d+e x)^2+1750 a A b^3 d e^2 (d+e x)-2205 a b^3 B d^3 e+2975 a b^3 B d^2 e (d+e x)-72 a b^3 B e (d+e x)^3-728 a b^3 B d e (d+e x)^2+525 A b^4 d^3 e-875 A b^4 d^2 e (d+e x)+40 A b^4 e (d+e x)^3+280 A b^4 d e (d+e x)^2+420 b^4 B d^4-700 b^4 B d^3 (d+e x)+224 b^4 B d^2 (d+e x)^2+24 b^4 B (d+e x)^4+32 b^4 B d (d+e x)^3\right )}{60 b^5 (a e+b (d+e x)-b d)^2}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 1060, normalized size = 2.60
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 685, normalized size = 1.68 \begin {gather*} \frac {7 \, {\left (4 \, B b^{3} d^{3} e^{2} - 17 \, B a b^{2} d^{2} e^{3} + 5 \, A b^{3} d^{2} e^{3} + 22 \, B a^{2} b d e^{4} - 10 \, A a b^{2} d e^{4} - 9 \, B a^{3} e^{5} + 5 \, A a^{2} b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {{\left (4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{2} - 4 \, \sqrt {x e + d} B b^{4} d^{4} e^{2} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{3} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{3} + 27 \, \sqrt {x e + d} B a b^{3} d^{3} e^{3} - 11 \, \sqrt {x e + d} A b^{4} d^{3} e^{3} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{4} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{4} - 57 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{4} + 33 \, \sqrt {x e + d} A a b^{3} d^{2} e^{4} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{5} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{5} + 49 \, \sqrt {x e + d} B a^{3} b d e^{5} - 33 \, \sqrt {x e + d} A a^{2} b^{2} d e^{5} - 15 \, \sqrt {x e + d} B a^{4} e^{6} + 11 \, \sqrt {x e + d} A a^{3} b e^{6}\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} e^{6} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d e^{6} + 45 \, \sqrt {x e + d} B b^{12} d^{2} e^{6} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} e^{7} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} e^{7} - 135 \, \sqrt {x e + d} B a b^{11} d e^{7} + 45 \, \sqrt {x e + d} A b^{12} d e^{7} + 90 \, \sqrt {x e + d} B a^{2} b^{10} e^{8} - 45 \, \sqrt {x e + d} A a b^{11} e^{8}\right )} e^{\left (-5\right )}}{15 \, b^{15} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 1873, normalized size = 4.60
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\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.
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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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