Optimal. Leaf size=424 \[ -\frac {(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}}+\frac {35 e^3 (a+b x) \sqrt {d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.35, antiderivative size = 424, 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 {35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {35 e^3 (a+b x) \sqrt {d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}}-\frac {(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \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+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \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 )^{5/2}} \, dx &=\frac {\left (b^4 \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 )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d+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 )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e (8 b B d+A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 (8 b B d+A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{192 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 (8 b B d+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}{128 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 \left (b^2 d-a b e\right ) (8 b B d+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}{128 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right ) (8 b B d+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 )}{64 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+A b e-9 a B e) (a+b x) \sqrt {d+e x}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (8 b B d+A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (8 b B d+A b e-9 a B e) (d+e x)^{5/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+A b e-9 a B e) (d+e x)^{7/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d+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 )}{64 b^{11/2} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 114, normalized size = 0.27 \begin {gather*} \frac {(d+e x)^{9/2} \left (\frac {e^3 (a+b x)^4 (-9 a B e+A b e+8 b B d) \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+9 a B-9 A b\right )}{36 b (a+b x)^3 \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 64.73, size = 502, normalized size = 1.18 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e^3 \sqrt {d+e x} \left (-945 a^4 B e^4+105 a^3 A b e^4-3465 a^3 b B e^3 (d+e x)+3675 a^3 b B d e^3+385 a^2 A b^2 e^3 (d+e x)-315 a^2 A b^2 d e^3-5355 a^2 b^2 B d^2 e^2-4599 a^2 b^2 B e^2 (d+e x)^2+10010 a^2 b^2 B d e^2 (d+e x)+315 a A b^3 d^2 e^2+511 a A b^3 e^2 (d+e x)^2-770 a A b^3 d e^2 (d+e x)+3465 a b^3 B d^3 e-9625 a b^3 B d^2 e (d+e x)-2511 a b^3 B e (d+e x)^3+8687 a b^3 B d e (d+e x)^2-105 A b^4 d^3 e+385 A b^4 d^2 e (d+e x)+279 A b^4 e (d+e x)^3-511 A b^4 d e (d+e x)^2-840 b^4 B d^4+3080 b^4 B d^3 (d+e x)-4088 b^4 B d^2 (d+e x)^2-384 b^4 B (d+e x)^4+2232 b^4 B d (d+e x)^3\right )}{192 b^5 (a e+b (d+e x)-b d)^4}+\frac {35 \left (-9 a B e^4+A b e^4+8 b B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{11/2} \sqrt {a e-b d}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1485, normalized size = 3.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 620, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{3}}{b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {35 \, {\left (8 \, B b d e^{3} - 9 \, B a e^{4} + A b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \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 {696 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 1784 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 1544 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 456 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} - 975 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} + 279 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 4079 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} - 511 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 5017 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 1929 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} - 105 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 5402 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} - 770 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 3051 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} + 315 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} + 385 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} + 2139 \, \sqrt {x e + d} B a^{3} b d e^{6} - 315 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} - 561 \, \sqrt {x e + d} B a^{4} e^{7} + 105 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5} \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 = 0.08, size = 1390, normalized size = 3.28
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 {5}{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 )}^{5/2}} \,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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