Optimal. Leaf size=111 \[ -\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 65, 223,
212} \begin {gather*} -\frac {2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}}-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {B \int \frac {\sqrt {d+e x}}{(a+b x)^{3/2}} \, dx}{b}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(B e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b^2}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 B e) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 B e) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^3}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 116, normalized size = 1.05 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (A b^2 (d+e x)+B \left (-3 a^2 e+3 b^2 d x+2 a b (d-2 e x)\right )\right )}{3 b^2 (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs.
\(2(89)=178\).
time = 0.23, size = 503, normalized size = 4.53
method | result | size |
default | \(\frac {\left (3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} e^{2} x^{2}-3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d e \,x^{2}+6 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2} x -6 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d e x +2 A \,b^{2} e x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} e^{2}-3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b d e -8 B a b e x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+6 B \,b^{2} d x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+2 A \,b^{2} d \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-6 B \,a^{2} e \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+4 B a b d \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\right ) \sqrt {e x +d}}{3 \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} \left (b x +a \right )^{\frac {3}{2}}}\) | \(503\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (91) = 182\).
time = 1.38, size = 524, normalized size = 4.72 \begin {gather*} \left [\frac {\frac {3 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d - {\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} e\right )} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + \frac {4 \, {\left (b^{2} d + {\left (2 \, b^{2} x + a b\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} e^{\frac {1}{2}}}{\sqrt {b}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right )}{\sqrt {b}} - 4 \, {\left (3 \, B b^{2} d x + {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B a^{2} + {\left (4 \, B a b - A b^{2}\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{6 \, {\left (b^{5} d x^{2} + 2 \, a b^{4} d x + a^{2} b^{3} d - {\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} e\right )}}, -\frac {3 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d - {\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} e\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {-\frac {e}{b}}}{2 \, {\left ({\left (b x^{2} + a x\right )} e^{2} + {\left (b d x + a d\right )} e\right )}}\right ) + 2 \, {\left (3 \, B b^{2} d x + {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B a^{2} + {\left (4 \, B a b - A b^{2}\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{3 \, {\left (b^{5} d x^{2} + 2 \, a b^{4} d x + a^{2} b^{3} d - {\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 522 vs.
\(2 (91) = 182\).
time = 0.64, size = 522, normalized size = 4.70 \begin {gather*} -\frac {B {\left | b \right |} e^{\frac {1}{2}} \log \left ({\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{b^{\frac {7}{2}}} - \frac {4 \, {\left (3 \, B b^{\frac {11}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 10 \, B a b^{\frac {9}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + A b^{\frac {11}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} + 11 \, B a^{2} b^{\frac {7}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 2 \, A a b^{\frac {9}{2}} d {\left | b \right |} e^{\frac {5}{2}} + 12 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {1}{2}} - 4 \, B a^{3} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {7}{2}} + A a^{2} b^{\frac {7}{2}} {\left | b \right |} e^{\frac {7}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B a \sqrt {b} {\left | b \right |} e^{\frac {3}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{\frac {3}{2}} {\left | b \right |} e^{\frac {3}{2}}\right )}}{3 \, {\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3} b^{3}} \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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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