Optimal. Leaf size=201 \[ \frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 52, 65,
223, 212} \begin {gather*} \frac {\sqrt {e} (-5 a B e+2 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}}+\frac {e \sqrt {a+b x} \sqrt {d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac {2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(3 b B d+2 A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{b^2 (b d-a e)}\\ &=\frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 b^3}\\ &=\frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a 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^4}\\ &=\frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(e (3 b B d+2 A b e-5 a 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^4}\\ &=\frac {e (3 b B d+2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{b^3 (b d-a e)}-\frac {2 (3 b B d+2 A b e-5 a B e) (d+e x)^{3/2}}{3 b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 133, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {d+e x} \left (B \left (-15 a^2 e+4 a b (d-5 e x)-3 b^2 x (-2 d+e x)\right )+2 A b (3 a e+b (d+4 e x))\right )}{3 b^3 (a+b x)^{3/2}}+\frac {\sqrt {e} (3 b B d+2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2}} \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. \(697\) vs.
\(2(173)=346\).
time = 0.29, size = 698, normalized size = 3.47
method | result | size |
default | \(\frac {\sqrt {e x +d}\, \left (6 A \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} e^{2} x^{2}-15 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}+9 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}+12 A \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 -30 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 +18 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 +6 B \,b^{2} e \,x^{2} \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+6 A \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}-16 A \,b^{2} e x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-15 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}+9 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 +40 B a b e x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-12 B \,b^{2} d x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-12 A a b e \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-4 A \,b^{2} d \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+30 B \,a^{2} e \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-8 B a b d \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\right )}{6 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} \left (b x +a \right )^{\frac {3}{2}}}\) | \(698\) |
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 [A]
time = 1.60, size = 550, normalized size = 2.74 \begin {gather*} \left [\frac {\frac {3 \, {\left (3 \, B b^{3} d x^{2} + 6 \, B a b^{2} d x + 3 \, B a^{2} b d - {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\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 (6 \, B b^{2} d x + 2 \, {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 6 \, A a b + 4 \, {\left (5 \, B a b - 2 \, A b^{2}\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{12 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {3 \, {\left (3 \, B b^{3} d x^{2} + 6 \, B a b^{2} d x + 3 \, B a^{2} b d - {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\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 (6 \, B b^{2} d x + 2 \, {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 6 \, A a b + 4 \, {\left (5 \, B a b - 2 \, A b^{2}\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{6 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\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 ) \left (d + e x\right )^{\frac {3}{2}}}{\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. 951 vs.
\(2 (185) = 370\).
time = 0.73, size = 951, normalized size = 4.73 \begin {gather*} \frac {\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} B {\left | b \right |} e}{b^{5}} - \frac {{\left (3 \, B b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {1}{2}} - 5 \, B a \sqrt {b} {\left | b \right |} e^{\frac {3}{2}} + 2 \, A b^{\frac {3}{2}} {\left | b \right |} e^{\frac {3}{2}}\right )} \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 )}{2 \, b^{5}} - \frac {4 \, {\left (3 \, B b^{\frac {13}{2}} d^{4} {\left | b \right |} e^{\frac {1}{2}} - 16 \, B a b^{\frac {11}{2}} d^{3} {\left | b \right |} e^{\frac {3}{2}} + 4 \, A b^{\frac {13}{2}} d^{3} {\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 {9}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} + 30 \, B a^{2} b^{\frac {9}{2}} d^{2} {\left | b \right |} e^{\frac {5}{2}} - 12 \, A a b^{\frac {11}{2}} d^{2} {\left | b \right |} e^{\frac {5}{2}} + 24 \, {\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 {7}{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} A b^{\frac {9}{2}} d^{2} {\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 {5}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} - 24 \, B a^{3} b^{\frac {7}{2}} d {\left | b \right |} e^{\frac {7}{2}} + 12 \, A a^{2} b^{\frac {9}{2}} d {\left | b \right |} e^{\frac {7}{2}} - 30 \, {\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 {5}{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} A a b^{\frac {7}{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 )}^{4} B a b^{\frac {3}{2}} d {\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 )}^{4} A b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 7 \, B a^{4} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {9}{2}} - 4 \, A a^{3} b^{\frac {7}{2}} {\left | b \right |} e^{\frac {9}{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^{3} b^{\frac {3}{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} A a^{2} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {7}{2}} + 9 \, {\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^{2} \sqrt {b} {\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} A a b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{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^{4}} \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 )}^{3/2}}{{\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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