Optimal. Leaf size=359 \[ -\frac {(d+e x)^{7/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e-A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {5 e^3 (a+b x) (-7 a B e-A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac {5 e^2 \sqrt {d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {5 e (d+e x)^{3/2} (-7 a B e-A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.32, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {770, 78, 47, 63, 208} \begin {gather*} -\frac {5 e^2 \sqrt {d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {5 e^3 (a+b x) (-7 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^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac {(d+e x)^{7/2} (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)^{5/2} (-7 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 {5 e (d+e x)^{3/2} (-7 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 63
Rule 78
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{\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)^{5/2}}{\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)^{7/2}}{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-7 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 )^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-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e (8 b B d-A b e-7 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 )^3} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e (8 b B d-A b e-7 a B e) (d+e x)^{3/2}}{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-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^2 (8 b B d-A b e-7 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (8 b B d-A b e-7 a B e) (d+e x)^{3/2}}{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-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^3 (8 b B d-A b e-7 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^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (8 b B d-A b e-7 a B e) (d+e x)^{3/2}}{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-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^2 (8 b B d-A b e-7 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^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^2 (8 b B d-A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (8 b B d-A b e-7 a B e) (d+e x)^{3/2}}{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-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^3 (8 b B d-A b e-7 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{9/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 219, normalized size = 0.61 \begin {gather*} \frac {-\frac {(a+b x) (-7 a B e-A b e+8 b B d) \left (b (d+e x) \sqrt {a e-b d} \left (15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left (8 d^2+26 d e x+33 e^2 x^2\right )\right )-15 \sqrt {b} e^3 (a+b x)^3 \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )\right )}{3 \sqrt {a e-b d}}-16 b^4 (d+e x)^4 (A b-a B)}{64 b^5 (a+b x)^3 \sqrt {(a+b x)^2} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 61.06, size = 511, normalized size = 1.42 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e^3 \sqrt {d+e x} \left (-105 a^4 B e^4-15 a^3 A b e^4-385 a^3 b B e^3 (d+e x)+435 a^3 b B d e^3-55 a^2 A b^2 e^3 (d+e x)+45 a^2 A b^2 d e^3-675 a^2 b^2 B d^2 e^2-511 a^2 b^2 B e^2 (d+e x)^2+1210 a^2 b^2 B d e^2 (d+e x)-45 a A b^3 d^2 e^2-73 a A b^3 e^2 (d+e x)^2+110 a A b^3 d e^2 (d+e x)+465 a b^3 B d^3 e-1265 a b^3 B d^2 e (d+e x)-279 a b^3 B e (d+e x)^3+1095 a b^3 B d e (d+e x)^2+15 A b^4 d^3 e-55 A b^4 d^2 e (d+e x)+15 A b^4 e (d+e x)^3+73 A b^4 d e (d+e x)^2-120 b^4 B d^4+440 b^4 B d^3 (d+e x)-584 b^4 B d^2 (d+e x)^2+264 b^4 B d (d+e x)^3\right )}{192 b^4 (b d-a e) (-a e-b (d+e x)+b d)^4}+\frac {5 \left (-7 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^{9/2} (b d-a e) \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 = 0.48, size = 1547, normalized size = 4.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 647, normalized size = 1.80 \begin {gather*} \frac {5 \, {\left (8 \, B b d e^{3} - 7 \, 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 \, {\left (b^{5} d \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a b^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {264 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 584 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 440 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 120 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} - 279 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} + 15 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 1095 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 73 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 1265 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} - 55 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 465 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} + 15 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} - 511 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 73 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 1210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} + 110 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 675 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} - 45 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} - 385 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} - 55 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} + 435 \, \sqrt {x e + d} B a^{3} b d e^{6} + 45 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} - 105 \, \sqrt {x e + d} B a^{4} e^{7} - 15 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left (b^{5} d \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a b^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1273, normalized size = 3.55
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 {5}{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 )}^{5/2}}{{\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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