Optimal. Leaf size=341 \[ -\frac {(d+e x)^{7/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e+3 A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {5 e (a+b x) \sqrt {b d-a e} (-7 a B e+3 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (a+b x) \sqrt {d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (a+b x) (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.30, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 7, 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)^{7/2} (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)^{5/2} (-7 a B e+3 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 {5 e (a+b x) (d+e x)^{3/2} (-7 a B e+3 A b e+4 b B d)}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {5 e (a+b x) \sqrt {d+e x} (-7 a B e+3 A b e+4 b B d)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (a+b x) \sqrt {b d-a e} (-7 a B e+3 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^{9/2} \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)^{5/2}}{\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)^{5/2}}{\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)^{7/2}}{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+3 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 )^2} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e (4 b B d+3 A b e-7 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^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) (d+e x)^{3/2}}{12 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e \left (b^2 d-a b e\right ) (4 b B d+3 A b e-7 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^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) (d+e x)^{3/2}}{12 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e \left (b^2 d-a b e\right )^2 (4 b B d+3 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}{8 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) (d+e x)^{3/2}}{12 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 \left (b^2 d-a b e\right )^2 (4 b B d+3 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 )}{4 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e (4 b B d+3 A b e-7 a B e) (a+b x) (d+e x)^{3/2}}{12 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(4 b B d+3 A b e-7 a B e) (d+e x)^{5/2}}{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)^{7/2}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e \sqrt {b d-a e} (4 b B d+3 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 )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 111, normalized size = 0.33 \begin {gather*} \frac {(a+b x) (d+e x)^{7/2} \left (\frac {e (a+b x)^2 (-7 a B e+3 A b e+4 b B d) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+7 a B-7 A b\right )}{14 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 = 59.94, size = 372, normalized size = 1.09 \begin {gather*} \frac {(-a e-b e x) \left (\frac {5 \left (7 a^2 B e^3-3 a A b e^3-11 a b B d e^2+3 A b^2 d e^2+4 b^2 B d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{9/2} \sqrt {a e-b d}}-\frac {e \sqrt {d+e x} \left (-105 a^3 B e^3+45 a^2 A b e^3-175 a^2 b B e^2 (d+e x)+270 a^2 b B d e^2+75 a A b^2 e^2 (d+e x)-90 a A b^2 d e^2-225 a b^2 B d^2 e-56 a b^2 B e (d+e x)^2+275 a b^2 B d e (d+e x)+45 A b^3 d^2 e+24 A b^3 e (d+e x)^2-75 A b^3 d e (d+e x)+60 b^3 B d^3-100 b^3 B d^2 (d+e x)+8 b^3 B (d+e x)^3+32 b^3 B d (d+e x)^2\right )}{12 b^4 (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 = 0.46, size = 680, normalized size = 1.99 \begin {gather*} \left [-\frac {15 \, {\left (4 \, B a^{2} b d e - {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (8 \, B b^{3} e^{2} x^{3} - 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \, {\left (7 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} - {\left (12 \, B b^{3} d^{2} - {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (4 \, B a^{2} b d e - {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, B b^{3} e^{2} x^{3} - 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{2} + 5 \, {\left (19 \, B a^{2} b - 3 \, A a b^{2}\right )} d e - 15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 8 \, {\left (7 \, B b^{3} d e - {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} e^{2}\right )} x^{2} - {\left (12 \, B b^{3} d^{2} - {\left (163 \, B a b^{2} - 27 \, A b^{3}\right )} d e + 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 478, normalized size = 1.40 \begin {gather*} \frac {5 \, {\left (4 \, B b^{2} d^{2} e^{2} - 11 \, B a b d e^{3} + 3 \, A b^{2} d e^{3} + 7 \, B a^{2} e^{4} - 3 \, A a b e^{4}\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^{4} \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^{3} d^{2} e^{2} - 4 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} + 19 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} - 7 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} - 26 \, \sqrt {x e + d} B a^{2} b d e^{4} + 14 \, \sqrt {x e + d} A a b^{2} d e^{4} + 11 \, \sqrt {x e + d} B a^{3} e^{5} - 7 \, \sqrt {x e + d} A a^{2} b e^{5}\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{6} e^{4} + 6 \, \sqrt {x e + d} B b^{6} d e^{4} - 9 \, \sqrt {x e + d} B a b^{5} e^{5} + 3 \, \sqrt {x e + d} A b^{6} e^{5}\right )} e^{\left (-3\right )}}{3 \, b^{9} \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 = 1150, normalized size = 3.37
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 {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 )}^{5/2}}{{\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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