Optimal. Leaf size=308 \[ -\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) \sqrt {d+e x} (b d-a e)^2}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 47, 50, 63, 208} \[ \frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) \sqrt {d+e x} (b d-a e)^2}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{8 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^2 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 67, normalized size = 0.22 \[ -\frac {2 e^2 (a+b x) (d+e x)^{11/2} \, _2F_1\left (3,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{11 \sqrt {(a+b x)^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 730, normalized size = 2.37 \[ \left [\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{40 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{20 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 446, normalized size = 1.45 \[ \frac {63 \, {\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {17 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{2} - 15 \, \sqrt {x e + d} b^{4} d^{4} e^{2} - 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{3} + 60 \, \sqrt {x e + d} a b^{3} d^{3} e^{3} + 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{4} - 90 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{4} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{5} + 60 \, \sqrt {x e + d} a^{3} b d e^{5} - 15 \, \sqrt {x e + d} a^{4} e^{6}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} b^{12} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{12} d e^{2} + 30 \, \sqrt {x e + d} b^{12} d^{2} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{11} e^{3} - 60 \, \sqrt {x e + d} a b^{11} d e^{3} + 30 \, \sqrt {x e + d} a^{2} b^{10} e^{4}\right )}}{5 \, b^{15} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1115, normalized size = 3.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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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