Optimal. Leaf size=232 \[ -\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {3 \sqrt {d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}}+\frac {3 \sqrt {d+e x} \left (x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+b d (4 c d-3 b e)\right )}{4 b^4 \left (b x+c x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {738, 820, 826, 1166, 208} \begin {gather*} \frac {3 \sqrt {d+e x} \left (x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+b d (4 c d-3 b e)\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {3 \sqrt {d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 738
Rule 820
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{2} d (4 c d-3 b e)+\frac {3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac {3}{4} e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} d e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )+\frac {3}{4} d e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac {3}{4} e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {\left (3 (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}+\frac {\left (3 c d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {3 \sqrt {d} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.41, size = 222, normalized size = 0.96 \begin {gather*} \frac {-3 \sqrt {d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {3 \sqrt {c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{\sqrt {c}}+\frac {b \sqrt {d+e x} \left (b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )+b^2 c x \left (8 d^2-37 d e x+3 e^2 x^2\right )+12 b c^2 d x^2 (3 d-2 e x)+24 c^3 d^2 x^3\right )}{x^2 (b+c x)^2}}{4 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.16, size = 392, normalized size = 1.69 \begin {gather*} -\frac {3 \left (5 b^2 \sqrt {d} e^2-20 b c d^{3/2} e+16 c^2 d^{5/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}-\frac {3 \sqrt {b e-c d} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 \sqrt {c}}+\frac {\sqrt {d+e x} \left (12 b^3 d^2 e^3-19 b^3 d e^3 (d+e x)+5 b^3 e^3 (d+e x)^2-48 b^2 c d^3 e^2+91 b^2 c d^2 e^2 (d+e x)-46 b^2 c d e^2 (d+e x)^2+3 b^2 c e^2 (d+e x)^3+60 b c^2 d^4 e-144 b c^2 d^3 e (d+e x)+108 b c^2 d^2 e (d+e x)^2-24 b c^2 d e (d+e x)^3-24 c^3 d^5+72 c^3 d^4 (d+e x)-72 c^3 d^3 (d+e x)^2+24 c^3 d^2 (d+e x)^3\right )}{4 b^4 e x^2 (b e+c (d+e x)-c d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 1661, normalized size = 7.16
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.25, size = 448, normalized size = 1.93 \begin {gather*} -\frac {3 \, {\left (16 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5}} + \frac {3 \, {\left (16 \, c^{2} d^{3} - 20 \, b c d^{2} e + 5 \, b^{2} d e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e - 24 \, \sqrt {x e + d} c^{3} d^{5} e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} d e^{2} + 108 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d^{2} e^{2} - 144 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{3} e^{2} + 60 \, \sqrt {x e + d} b c^{2} d^{4} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c e^{3} - 46 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c d e^{3} + 91 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d^{2} e^{3} - 48 \, \sqrt {x e + d} b^{2} c d^{3} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{4} - 19 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{4} + 12 \, \sqrt {x e + d} b^{3} d^{2} e^{4}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 521, normalized size = 2.25 \begin {gather*} \frac {5 \sqrt {e x +d}\, e^{4}}{4 \left (c e x +b e \right )^{2} b}-\frac {11 \sqrt {e x +d}\, c d \,e^{3}}{2 \left (c e x +b e \right )^{2} b^{2}}+\frac {29 \sqrt {e x +d}\, c^{2} d^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{3} d^{3} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c \,e^{3}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {3 e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {15 \left (e x +d \right )^{\frac {3}{2}} c^{2} d \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {39 c d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{2} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {21 c^{2} d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 c^{3} d^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{5}}-\frac {15 \sqrt {d}\, e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3}}+\frac {15 c \,d^{\frac {3}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {12 c^{2} d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}+\frac {7 \sqrt {e x +d}\, d^{2}}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c \,d^{3}}{b^{4} e \,x^{2}}-\frac {9 \left (e x +d \right )^{\frac {3}{2}} d}{4 b^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c \,d^{2}}{b^{4} e \,x^{2}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 0.58, size = 910, normalized size = 3.92 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\frac {81\,c^2\,d^2\,e^8\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{8\,\left (\frac {189\,c^3\,d^3\,e^8}{8}-\frac {351\,b\,c^2\,d^2\,e^9}{32}-\frac {27\,c^4\,d^4\,e^7}{2\,b}+\frac {27\,b^2\,c\,d\,e^{10}}{32}\right )}+\frac {27\,c^3\,d^3\,e^7\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,\left (-\frac {27\,b^3\,c\,d\,e^{10}}{32}+\frac {351\,b^2\,c^2\,d^2\,e^9}{32}-\frac {189\,b\,c^3\,d^3\,e^8}{8}+\frac {27\,c^4\,d^4\,e^7}{2}\right )}+\frac {27\,c\,d\,e^9\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{32\,\left (\frac {351\,c^2\,d^2\,e^9}{32}-\frac {27\,b\,c\,d\,e^{10}}{32}-\frac {189\,c^3\,d^3\,e^8}{8\,b}+\frac {27\,c^4\,d^4\,e^7}{2\,b^2}\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (b^2\,e^2-12\,b\,c\,d\,e+16\,c^2\,d^2\right )}{4\,b^5\,c}-\frac {3\,\sqrt {d}\,\mathrm {atanh}\left (\frac {135\,c\,\sqrt {d}\,e^{10}\,\sqrt {d+e\,x}}{32\,\left (\frac {135\,c\,d\,e^{10}}{32}-\frac {675\,c^2\,d^2\,e^9}{32\,b}+\frac {243\,c^3\,d^3\,e^8}{8\,b^2}-\frac {27\,c^4\,d^4\,e^7}{2\,b^3}\right )}+\frac {675\,c^2\,d^{3/2}\,e^9\,\sqrt {d+e\,x}}{32\,\left (\frac {675\,c^2\,d^2\,e^9}{32}-\frac {135\,b\,c\,d\,e^{10}}{32}-\frac {243\,c^3\,d^3\,e^8}{8\,b}+\frac {27\,c^4\,d^4\,e^7}{2\,b^2}\right )}+\frac {243\,c^3\,d^{5/2}\,e^8\,\sqrt {d+e\,x}}{8\,\left (\frac {243\,c^3\,d^3\,e^8}{8}-\frac {675\,b\,c^2\,d^2\,e^9}{32}-\frac {27\,c^4\,d^4\,e^7}{2\,b}+\frac {135\,b^2\,c\,d\,e^{10}}{32}\right )}+\frac {27\,c^4\,d^{7/2}\,e^7\,\sqrt {d+e\,x}}{2\,\left (-\frac {135\,b^3\,c\,d\,e^{10}}{32}+\frac {675\,b^2\,c^2\,d^2\,e^9}{32}-\frac {243\,b\,c^3\,d^3\,e^8}{8}+\frac {27\,c^4\,d^4\,e^7}{2}\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{4\,b^5}-\frac {\frac {{\left (d+e\,x\right )}^{3/2}\,\left (19\,b^3\,d\,e^4-91\,b^2\,c\,d^2\,e^3+144\,b\,c^2\,d^3\,e^2-72\,c^3\,d^4\,e\right )}{4\,b^4}+\frac {3\,\sqrt {d+e\,x}\,\left (-b^3\,d^2\,e^4+4\,b^2\,c\,d^3\,e^3-5\,b\,c^2\,d^4\,e^2+2\,c^3\,d^5\,e\right )}{b^4}-\frac {\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,b^2\,e^3-36\,b\,c\,d\,e^2+36\,c^2\,d^2\,e\right )}{4\,b^4}-\frac {3\,c\,e\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-8\,b\,c\,d\,e+8\,c^2\,d^2\right )}{4\,b^4}}{c^2\,{\left (d+e\,x\right )}^4-\left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )-\left (4\,c^2\,d-2\,b\,c\,e\right )\,{\left (d+e\,x\right )}^3+{\left (d+e\,x\right )}^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )+c^2\,d^4+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________