Optimal. Leaf size=285 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) (d+e x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}+\frac {3 b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^4 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)} \]
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Rubi [A] time = 0.27, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{e^5 (a+b x) (d+e x)}+\frac {3 b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^4 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
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)^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4}-\frac {b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^2}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)}{e^4}+\frac {b^6 B (d+e x)^2}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {3 b (b d-a e) (2 b B d-A b e-a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {(b d-a e)^3 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}+\frac {b^3 B (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 262, normalized size = 0.92 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (6 a^3 e^3 (B d-A e)+18 a^2 b e^2 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+9 a b^2 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )-6 (d+e x) (b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)+b^3 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )\right )}{6 e^5 (a+b x) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 5.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 396, normalized size = 1.39 \begin {gather*} \frac {2 \, B b^{3} e^{4} x^{4} - 6 \, B b^{3} d^{4} - 6 \, A a^{3} e^{4} + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - {\left (4 \, B b^{3} d e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (4 \, B b^{3} d^{2} e^{2} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (3 \, B b^{3} d^{3} e - 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x - 6 \, {\left (4 \, B b^{3} d^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + {\left (4 \, B b^{3} d^{3} e - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 425, normalized size = 1.49 \begin {gather*} -{\left (4 \, B b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, B a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, A b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, B b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, B b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, B a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, A b^{3} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 36 \, B a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) - 12 \, A b^{3} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, B a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{2} x e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} - \frac {{\left (B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) - A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 540, normalized size = 1.89 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 B \,b^{3} e^{4} x^{4}+3 A \,b^{3} e^{4} x^{3}+9 B a \,b^{2} e^{4} x^{3}-4 B \,b^{3} d \,e^{3} x^{3}+18 A \,a^{2} b \,e^{4} x \ln \left (e x +d \right )-36 A a \,b^{2} d \,e^{3} x \ln \left (e x +d \right )+18 A a \,b^{2} e^{4} x^{2}+18 A \,b^{3} d^{2} e^{2} x \ln \left (e x +d \right )-9 A \,b^{3} d \,e^{3} x^{2}+6 B \,a^{3} e^{4} x \ln \left (e x +d \right )-36 B \,a^{2} b d \,e^{3} x \ln \left (e x +d \right )+18 B \,a^{2} b \,e^{4} x^{2}+54 B a \,b^{2} d^{2} e^{2} x \ln \left (e x +d \right )-27 B a \,b^{2} d \,e^{3} x^{2}-24 B \,b^{3} d^{3} e x \ln \left (e x +d \right )+12 B \,b^{3} d^{2} e^{2} x^{2}+18 A \,a^{2} b d \,e^{3} \ln \left (e x +d \right )-36 A a \,b^{2} d^{2} e^{2} \ln \left (e x +d \right )+18 A a \,b^{2} d \,e^{3} x +18 A \,b^{3} d^{3} e \ln \left (e x +d \right )-12 A \,b^{3} d^{2} e^{2} x +6 B \,a^{3} d \,e^{3} \ln \left (e x +d \right )-36 B \,a^{2} b \,d^{2} e^{2} \ln \left (e x +d \right )+18 B \,a^{2} b d \,e^{3} x +54 B a \,b^{2} d^{3} e \ln \left (e x +d \right )-36 B a \,b^{2} d^{2} e^{2} x -24 B \,b^{3} d^{4} \ln \left (e x +d \right )+18 B \,b^{3} d^{3} e x -6 A \,a^{3} e^{4}+18 A \,a^{2} b d \,e^{3}-18 A a \,b^{2} d^{2} e^{2}+6 A \,b^{3} d^{3} e +6 B \,a^{3} d \,e^{3}-18 B \,a^{2} b \,d^{2} e^{2}+18 B a \,b^{2} d^{3} e -6 B \,b^{3} d^{4}\right )}{6 \left (b x +a \right )^{3} \left (e x +d \right ) e^{5}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \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 )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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