Optimal. Leaf size=260 \[ -\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) \sqrt {d+e x}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^5 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^5 (a+b x)}-\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) \sqrt {d+e x}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{5 e^5 (a+b x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^{7/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^{7/2}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{7/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{5/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^{3/2}}-\frac {4 b^3 (b d-a e)}{e^4 \sqrt {d+e x}}+\frac {b^4 \sqrt {d+e x}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac {8 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac {12 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}-\frac {8 b^3 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 172, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (3 a^4 e^4+4 a^3 b e^3 (2 d+5 e x)+6 a^2 b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-12 a b^3 e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (a+b x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 23.23, size = 241, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-3 a^4 e^4-20 a^3 b e^3 (d+e x)+12 a^3 b d e^3-18 a^2 b^2 d^2 e^2-90 a^2 b^2 e^2 (d+e x)^2+60 a^2 b^2 d e^2 (d+e x)+12 a b^3 d^3 e-60 a b^3 d^2 e (d+e x)+60 a b^3 e (d+e x)^3+180 a b^3 d e (d+e x)^2-3 b^4 d^4+20 b^4 d^3 (d+e x)-90 b^4 d^2 (d+e x)^2+5 b^4 (d+e x)^4-60 b^4 d (d+e x)^3\right )}{15 e^4 (d+e x)^{5/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 213, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e - 48 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 20 \, {\left (2 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} - 30 \, {\left (8 \, b^{4} d^{2} e^{2} - 12 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} - 20 \, {\left (16 \, b^{4} d^{3} e - 24 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 316, normalized size = 1.22 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{10} \mathrm {sgn}\left (b x + a\right ) - 12 \, \sqrt {x e + d} b^{4} d e^{10} \mathrm {sgn}\left (b x + a\right ) + 12 \, \sqrt {x e + d} a b^{3} e^{11} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, {\left (x e + d\right )} b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 180 \, {\left (x e + d\right )}^{2} a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + 60 \, {\left (x e + d\right )} a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 90 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 60 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, {\left (x e + d\right )} a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 202, normalized size = 0.78 \begin {gather*} -\frac {2 \left (-5 b^{4} e^{4} x^{4}-60 a \,b^{3} e^{4} x^{3}+40 b^{4} d \,e^{3} x^{3}+90 a^{2} b^{2} e^{4} x^{2}-360 a \,b^{3} d \,e^{3} x^{2}+240 b^{4} d^{2} e^{2} x^{2}+20 a^{3} b \,e^{4} x +120 a^{2} b^{2} d \,e^{3} x -480 a \,b^{3} d^{2} e^{2} x +320 b^{4} d^{3} e x +3 a^{4} e^{4}+8 a^{3} b d \,e^{3}+48 a^{2} b^{2} d^{2} e^{2}-192 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 326, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \, {\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} a}{5 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 144 \, a b^{2} d^{3} e - 24 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} - 5 \, {\left (8 \, b^{3} d e^{3} - 9 \, a b^{2} e^{4}\right )} x^{3} - 15 \, {\left (16 \, b^{3} d^{2} e^{2} - 18 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} - 5 \, {\left (64 \, b^{3} d^{3} e - 72 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} b}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 283, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {6\,a^4\,e^4+16\,a^3\,b\,d\,e^3+96\,a^2\,b^2\,d^2\,e^2-384\,a\,b^3\,d^3\,e+256\,b^4\,d^4}{15\,b\,e^7}-\frac {2\,b^3\,x^4}{3\,e^3}+\frac {x\,\left (40\,a^3\,b\,e^4+240\,a^2\,b^2\,d\,e^3-960\,a\,b^3\,d^2\,e^2+640\,b^4\,d^3\,e\right )}{15\,b\,e^7}-\frac {8\,b^2\,x^3\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^4}+\frac {4\,b\,x^2\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^5}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (15\,a\,e^7+30\,b\,d\,e^6\right )\,\sqrt {d+e\,x}}{15\,b\,e^7}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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