Optimal. Leaf size=281 \[ \frac {7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2}}-\frac {7 b e (5 a B e-9 A b e+4 b B d)}{4 \sqrt {d+e x} (b d-a e)^5}-\frac {7 e (5 a B e-9 A b e+4 b B d)}{12 (d+e x)^{3/2} (b d-a e)^4}-\frac {7 e (5 a B e-9 A b e+4 b B d)}{20 b (d+e x)^{5/2} (b d-a e)^3}-\frac {5 a B e-9 A b e+4 b B d}{4 b (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{2 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.27, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \[ \frac {7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2}}-\frac {7 b e (5 a B e-9 A b e+4 b B d)}{4 \sqrt {d+e x} (b d-a e)^5}-\frac {7 e (5 a B e-9 A b e+4 b B d)}{12 (d+e x)^{3/2} (b d-a e)^4}-\frac {7 e (5 a B e-9 A b e+4 b B d)}{20 b (d+e x)^{5/2} (b d-a e)^3}-\frac {5 a B e-9 A b e+4 b B d}{4 b (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{2 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^3 (d+e x)^{7/2}} \, dx &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}+\frac {(4 b B d-9 A b e+5 a B e) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{4 b (b d-a e)}\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {(7 e (4 b B d-9 A b e+5 a B e)) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {7 e (4 b B d-9 A b e+5 a B e)}{20 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {(7 e (4 b B d-9 A b e+5 a B e)) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^3}\\ &=-\frac {7 e (4 b B d-9 A b e+5 a B e)}{20 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {7 e (4 b B d-9 A b e+5 a B e)}{12 (b d-a e)^4 (d+e x)^{3/2}}-\frac {(7 b e (4 b B d-9 A b e+5 a B e)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^4}\\ &=-\frac {7 e (4 b B d-9 A b e+5 a B e)}{20 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {7 e (4 b B d-9 A b e+5 a B e)}{12 (b d-a e)^4 (d+e x)^{3/2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e)}{4 (b d-a e)^5 \sqrt {d+e x}}-\frac {\left (7 b^2 e (4 b B d-9 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^5}\\ &=-\frac {7 e (4 b B d-9 A b e+5 a B e)}{20 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {7 e (4 b B d-9 A b e+5 a B e)}{12 (b d-a e)^4 (d+e x)^{3/2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e)}{4 (b d-a e)^5 \sqrt {d+e x}}-\frac {\left (7 b^2 (4 b B d-9 A b e+5 a B e)\right ) \operatorname {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)^5}\\ &=-\frac {7 e (4 b B d-9 A b e+5 a B e)}{20 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{5/2}}-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{5/2}}-\frac {7 e (4 b B d-9 A b e+5 a B e)}{12 (b d-a e)^4 (d+e x)^{3/2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e)}{4 (b d-a e)^5 \sqrt {d+e x}}+\frac {7 b^{3/2} e (4 b B d-9 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 97, normalized size = 0.35 \[ \frac {\frac {e (-5 a B e+9 A b e-4 b B d) \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac {5 a B-5 A b}{(a+b x)^2}}{10 b (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 2675, normalized size = 9.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.42, size = 609, normalized size = 2.17 \[ -\frac {7 \, {\left (4 \, B b^{3} d e + 5 \, B a b^{2} e^{2} - 9 \, A b^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e - 4 \, \sqrt {x e + d} B b^{4} d^{2} e + 11 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} e^{2} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{2} - 9 \, \sqrt {x e + d} B a b^{3} d e^{2} + 17 \, \sqrt {x e + d} A b^{4} d e^{2} + 13 \, \sqrt {x e + d} B a^{2} b^{2} e^{3} - 17 \, \sqrt {x e + d} A a b^{3} e^{3}}{4 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B b^{2} d e + 10 \, {\left (x e + d\right )} B b^{2} d^{2} e + 3 \, B b^{2} d^{3} e + 45 \, {\left (x e + d\right )}^{2} B a b e^{2} - 90 \, {\left (x e + d\right )}^{2} A b^{2} e^{2} - 5 \, {\left (x e + d\right )} B a b d e^{2} - 15 \, {\left (x e + d\right )} A b^{2} d e^{2} - 6 \, B a b d^{2} e^{2} - 3 \, A b^{2} d^{2} e^{2} - 5 \, {\left (x e + d\right )} B a^{2} e^{3} + 15 \, {\left (x e + d\right )} A a b e^{3} + 3 \, B a^{2} d e^{3} + 6 \, A a b d e^{3} - 3 \, A a^{2} e^{4}\right )}}{15 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 648, normalized size = 2.31 \[ -\frac {17 \sqrt {e x +d}\, A a \,b^{3} e^{3}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}+\frac {17 \sqrt {e x +d}\, A \,b^{4} d \,e^{2}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}+\frac {13 \sqrt {e x +d}\, B \,a^{2} b^{2} e^{3}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}-\frac {9 \sqrt {e x +d}\, B a \,b^{3} d \,e^{2}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}-\frac {\sqrt {e x +d}\, B \,b^{4} d^{2} e}{\left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}-\frac {15 \left (e x +d \right )^{\frac {3}{2}} A \,b^{4} e^{2}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}-\frac {63 A \,b^{3} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {11 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} e^{2}}{4 \left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}+\frac {35 B a \,b^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {\left (e x +d \right )^{\frac {3}{2}} B \,b^{4} d e}{\left (a e -b d \right )^{5} \left (b x e +a e \right )^{2}}+\frac {7 B \,b^{3} d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {12 A \,b^{2} e^{2}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}+\frac {6 B a b \,e^{2}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}+\frac {6 B \,b^{2} d e}{\left (a e -b d \right )^{5} \sqrt {e x +d}}+\frac {2 A b \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 B a \,e^{2}}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 B b d e}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A \,e^{2}}{5 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 B d e}{5 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {5}{2}}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 418, normalized size = 1.49 \[ \frac {\frac {35\,{\left (d+e\,x\right )}^3\,\left (-9\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+5\,B\,a\,b^2\,e^2\right )}{12\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,\left (A\,e^2-B\,d\,e\right )}{5\,\left (a\,e-b\,d\right )}-\frac {2\,\left (d+e\,x\right )\,\left (5\,B\,a\,e^2-9\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,b^3\,{\left (d+e\,x\right )}^4\,\left (5\,B\,a\,e^2-9\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{4\,{\left (a\,e-b\,d\right )}^5}+\frac {14\,b\,{\left (d+e\,x\right )}^2\,\left (5\,B\,a\,e^2-9\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{15\,{\left (a\,e-b\,d\right )}^3}}{b^2\,{\left (d+e\,x\right )}^{9/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{7/2}+{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {7\,b^{3/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (5\,B\,a\,e-9\,A\,b\,e+4\,B\,b\,d\right )\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}\,\left (5\,B\,a\,e^2-9\,A\,b\,e^2+4\,B\,b\,d\,e\right )}\right )\,\left (5\,B\,a\,e-9\,A\,b\,e+4\,B\,b\,d\right )}{4\,{\left (a\,e-b\,d\right )}^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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