Optimal. Leaf size=197 \[ -\frac {3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt {d+e x}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}-\frac {4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt {d+e x}}+\frac {3 e (4 b B d-5 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65,
214} \begin {gather*} -\frac {A b-a B}{2 b (a+b x)^2 \sqrt {d+e x} (b d-a e)}-\frac {3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt {d+e x} (b d-a e)^2}+\frac {3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^3 (d+e x)^{3/2}} \, dx &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}+\frac {(4 b B d-5 A b e+a B e) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{4 b (b d-a e)}\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}-\frac {4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt {d+e x}}-\frac {(3 e (4 b B d-5 A b e+a B e)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt {d+e x}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}-\frac {4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt {d+e x}}-\frac {(3 e (4 b B d-5 A b e+a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^3}\\ &=-\frac {3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt {d+e x}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}-\frac {4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt {d+e x}}-\frac {(3 (4 b B d-5 A b e+a B e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 (b d-a e)^3}\\ &=-\frac {3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt {d+e x}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt {d+e x}}-\frac {4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt {d+e x}}+\frac {3 e (4 b B d-5 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.19, size = 194, normalized size = 0.98 \begin {gather*} \frac {1}{4} \left (\frac {-B \left (4 b^2 d x (d+3 e x)+a^2 e (13 d+5 e x)+a b \left (2 d^2+21 d e x+3 e^2 x^2\right )\right )+A \left (8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )\right )}{(b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {3 e (4 b B d-5 A b e+a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{7/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 195, normalized size = 0.99
method | result | size |
derivativedivides | \(2 e \left (-\frac {A e -B d}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {\frac {\left (\frac {7}{8} A \,b^{2} e -\frac {3}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {9}{8} A a b \,e^{2}-\frac {9}{8} A \,b^{2} d e -\frac {5}{8} B \,a^{2} e^{2}+\frac {1}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {3 \left (5 A b e -B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{\left (a e -b d \right )^{3}}\right )\) | \(195\) |
default | \(2 e \left (-\frac {A e -B d}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {\frac {\left (\frac {7}{8} A \,b^{2} e -\frac {3}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {9}{8} A a b \,e^{2}-\frac {9}{8} A \,b^{2} d e -\frac {5}{8} B \,a^{2} e^{2}+\frac {1}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {3 \left (5 A b e -B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}}{\left (a e -b d \right )^{3}}\right )\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 690 vs.
\(2 (190) = 380\).
time = 1.25, size = 1395, normalized size = 7.08 \begin {gather*} \left [\frac {3 \, \sqrt {b^{2} d - a b e} {\left ({\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} e^{3} + {\left (4 \, B b^{3} d x^{3} + {\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d x^{2} + 2 \, {\left (3 \, B a^{2} b - 5 \, A a b^{2}\right )} d x + {\left (B a^{3} - 5 \, A a^{2} b\right )} d\right )} e^{2} + 4 \, {\left (B b^{3} d^{2} x^{2} + 2 \, B a b^{2} d^{2} x + B a^{2} b d^{2}\right )} e\right )} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e + 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (4 \, B b^{4} d^{3} x + 2 \, {\left (B a b^{3} + A b^{4}\right )} d^{3} + {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - {\left (3 \, {\left (3 \, B a b^{3} + 5 \, A b^{4}\right )} d x^{2} + 4 \, {\left (4 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} d x + {\left (13 \, B a^{3} b - A a^{2} b^{2}\right )} d\right )} e^{2} + {\left (12 \, B b^{4} d^{2} x^{2} + {\left (17 \, B a b^{3} - 5 \, A b^{4}\right )} d^{2} x + 11 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{7} d^{5} x^{2} + 2 \, a b^{6} d^{5} x + a^{2} b^{5} d^{5} + {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} e^{5} - {\left (4 \, a^{3} b^{4} d x^{3} + 7 \, a^{4} b^{3} d x^{2} + 2 \, a^{5} b^{2} d x - a^{6} b d\right )} e^{4} + 2 \, {\left (3 \, a^{2} b^{5} d^{2} x^{3} + 4 \, a^{3} b^{4} d^{2} x^{2} - a^{4} b^{3} d^{2} x - 2 \, a^{5} b^{2} d^{2}\right )} e^{3} - 2 \, {\left (2 \, a b^{6} d^{3} x^{3} + a^{2} b^{5} d^{3} x^{2} - 4 \, a^{3} b^{4} d^{3} x - 3 \, a^{4} b^{3} d^{3}\right )} e^{2} + {\left (b^{7} d^{4} x^{3} - 2 \, a b^{6} d^{4} x^{2} - 7 \, a^{2} b^{5} d^{4} x - 4 \, a^{3} b^{4} d^{4}\right )} e\right )}}, -\frac {3 \, \sqrt {-b^{2} d + a b e} {\left ({\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} e^{3} + {\left (4 \, B b^{3} d x^{3} + {\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d x^{2} + 2 \, {\left (3 \, B a^{2} b - 5 \, A a b^{2}\right )} d x + {\left (B a^{3} - 5 \, A a^{2} b\right )} d\right )} e^{2} + 4 \, {\left (B b^{3} d^{2} x^{2} + 2 \, B a b^{2} d^{2} x + B a^{2} b d^{2}\right )} e\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) + {\left (4 \, B b^{4} d^{3} x + 2 \, {\left (B a b^{3} + A b^{4}\right )} d^{3} + {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - {\left (3 \, {\left (3 \, B a b^{3} + 5 \, A b^{4}\right )} d x^{2} + 4 \, {\left (4 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} d x + {\left (13 \, B a^{3} b - A a^{2} b^{2}\right )} d\right )} e^{2} + {\left (12 \, B b^{4} d^{2} x^{2} + {\left (17 \, B a b^{3} - 5 \, A b^{4}\right )} d^{2} x + 11 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{7} d^{5} x^{2} + 2 \, a b^{6} d^{5} x + a^{2} b^{5} d^{5} + {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} e^{5} - {\left (4 \, a^{3} b^{4} d x^{3} + 7 \, a^{4} b^{3} d x^{2} + 2 \, a^{5} b^{2} d x - a^{6} b d\right )} e^{4} + 2 \, {\left (3 \, a^{2} b^{5} d^{2} x^{3} + 4 \, a^{3} b^{4} d^{2} x^{2} - a^{4} b^{3} d^{2} x - 2 \, a^{5} b^{2} d^{2}\right )} e^{3} - 2 \, {\left (2 \, a b^{6} d^{3} x^{3} + a^{2} b^{5} d^{3} x^{2} - 4 \, a^{3} b^{4} d^{3} x - 3 \, a^{4} b^{3} d^{3}\right )} e^{2} + {\left (b^{7} d^{4} x^{3} - 2 \, a b^{6} d^{4} x^{2} - 7 \, a^{2} b^{5} d^{4} x - 4 \, a^{3} b^{4} d^{4}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.96, size = 346, normalized size = 1.76 \begin {gather*} -\frac {3 \, {\left (4 \, B b d e + B a e^{2} - 5 \, A b e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (B d e - A e^{2}\right )}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {x e + d}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {x e + d} B b^{2} d^{2} e + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{2} - 7 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{2} - \sqrt {x e + d} B a b d e^{2} + 9 \, \sqrt {x e + d} A b^{2} d e^{2} + 5 \, \sqrt {x e + d} B a^{2} e^{3} - 9 \, \sqrt {x e + d} A a b e^{3}}{4 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.46, size = 296, normalized size = 1.50 \begin {gather*} \frac {\frac {5\,\left (d+e\,x\right )\,\left (B\,a\,e^2-5\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{4\,{\left (a\,e-b\,d\right )}^2}-\frac {2\,\left (A\,e^2-B\,d\,e\right )}{a\,e-b\,d}+\frac {3\,b\,{\left (d+e\,x\right )}^2\,\left (B\,a\,e^2-5\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{4\,{\left (a\,e-b\,d\right )}^3}}{b^2\,{\left (d+e\,x\right )}^{5/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{3/2}+\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {3\,e\,\mathrm {atan}\left (\frac {3\,\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (B\,a\,e-5\,A\,b\,e+4\,B\,b\,d\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}\,\left (3\,B\,a\,e^2-15\,A\,b\,e^2+12\,B\,b\,d\,e\right )}\right )\,\left (B\,a\,e-5\,A\,b\,e+4\,B\,b\,d\right )}{4\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________