Optimal. Leaf size=130 \[ \frac {2 (A b-a B) (b d-a e) \sqrt {d+e x}}{b^3}+\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e}-\frac {2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214}
\begin {gather*} -\frac {2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)}{b^3}+\frac {2 (d+e x)^{3/2} (A b-a B)}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 214
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{a+b x} \, dx &=\frac {2 B (d+e x)^{5/2}}{5 b e}+\frac {\left (2 \left (\frac {5 A b e}{2}-\frac {5 a B e}{2}\right )\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{5 b e}\\ &=\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^2}\\ &=\frac {2 (A b-a B) (b d-a e) \sqrt {d+e x}}{b^3}+\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^3}\\ &=\frac {2 (A b-a B) (b d-a e) \sqrt {d+e x}}{b^3}+\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e}+\frac {\left (2 (A b-a B) (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^3 e}\\ &=\frac {2 (A b-a B) (b d-a e) \sqrt {d+e x}}{b^3}+\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac {2 B (d+e x)^{5/2}}{5 b e}-\frac {2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 129, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15 a^2 B e^2-5 a b e (4 B d+3 A e+B e x)+b^2 \left (3 B (d+e x)^2+5 A e (4 d+e x)\right )\right )}{15 b^3 e}+\frac {2 (A b-a B) (-b d+a e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 205, normalized size = 1.58
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+A a b \,e^{2} \sqrt {e x +d}-A \,b^{2} d e \sqrt {e x +d}-B \,a^{2} e^{2} \sqrt {e x +d}+B a b d e \sqrt {e x +d}\right )}{b^{3}}+\frac {2 e \left (A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-B \,a^{3} e^{2}+2 B \,a^{2} b d e -B a \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{3} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(205\) |
default | \(\frac {-\frac {2 \left (-\frac {b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+A a b \,e^{2} \sqrt {e x +d}-A \,b^{2} d e \sqrt {e x +d}-B \,a^{2} e^{2} \sqrt {e x +d}+B a b d e \sqrt {e x +d}\right )}{b^{3}}+\frac {2 e \left (A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-B \,a^{3} e^{2}+2 B \,a^{2} b d e -B a \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{3} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(205\) |
risch | \(-\frac {2 \left (-3 b^{2} B \,x^{2} e^{2}-5 A \,b^{2} e^{2} x +5 B a b \,e^{2} x -6 B \,b^{2} d e x +15 A a b \,e^{2}-20 A \,b^{2} d e -15 B \,a^{2} e^{2}+20 B a b d e -3 b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{15 e \,b^{3}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,a^{2} e^{2}}{b^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {4 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A a d e}{b \sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,d^{2}}{\sqrt {\left (a e -b d \right ) b}}-\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{3} e^{2}}{b^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {4 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{2} d e}{b^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a \,d^{2}}{b \sqrt {\left (a e -b d \right ) b}}\) | \(363\) |
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 [A]
time = 1.19, size = 372, normalized size = 2.86 \begin {gather*} \left [-\frac {{\left (15 \, {\left ({\left (B a b - A b^{2}\right )} d e - {\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (3 \, B b^{2} d^{2} + {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} e^{2} + 2 \, {\left (3 \, B b^{2} d x - 10 \, {\left (B a b - A b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{15 \, b^{3}}, \frac {2 \, {\left (15 \, {\left ({\left (B a b - A b^{2}\right )} d e - {\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (3 \, B b^{2} d^{2} + {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} e^{2} + 2 \, {\left (3 \, B b^{2} d x - 10 \, {\left (B a b - A b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{15 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 21.36, size = 139, normalized size = 1.07 \begin {gather*} \frac {2 B \left (d + e x\right )^{\frac {5}{2}}}{5 b e} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A b - 2 B a\right )}{3 b^{2}} + \frac {\sqrt {d + e x} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{b^{3}} - \frac {2 \left (- A b + B a\right ) \left (a e - b d\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{4} \sqrt {\frac {a e - b d}{b}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.66, size = 228, normalized size = 1.75 \begin {gather*} -\frac {2 \, {\left (B a b^{2} d^{2} - A b^{3} d^{2} - 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + B a^{3} e^{2} - A a^{2} b e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{3}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} e^{4} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} e^{5} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{5} - 15 \, \sqrt {x e + d} B a b^{3} d e^{5} + 15 \, \sqrt {x e + d} A b^{4} d e^{5} + 15 \, \sqrt {x e + d} B a^{2} b^{2} e^{6} - 15 \, \sqrt {x e + d} A a b^{3} e^{6}\right )} e^{\left (-5\right )}}{15 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 236, normalized size = 1.82 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{3\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{3\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{-B\,a^3\,e^2+2\,B\,a^2\,b\,d\,e+A\,a^2\,b\,e^2-B\,a\,b^2\,d^2-2\,A\,a\,b^2\,d\,e+A\,b^3\,d^2}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{b^{7/2}}+\frac {2\,B\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,e}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,\sqrt {d+e\,x}}{b\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________