Optimal. Leaf size=236 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) \log (d+e x)}{e^5 (a+b x)}-\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{e^4 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{3 e^2}+\frac {B (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b e} \]
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Rubi [A] time = 0.17, antiderivative size = 236, 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 {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{e^4 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{3 e^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) \log (d+e x)}{e^5 (a+b x)}+\frac {B (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b e} \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} \, 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} \, 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 {b^4 (b d-a e)^2 (-B d+A e)}{e^4}-\frac {b^4 (b d-a e) (-B d+A e) (a+b x)}{e^3}+\frac {b^4 (-B d+A e) (a+b x)^2}{e^2}+\frac {B \left (a b+b^2 x\right )^3}{e}-\frac {b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {b (b d-a e)^2 (B d-A e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {(b d-a e) (B d-A e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3}-\frac {(B d-A e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2}+\frac {B (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b e}+\frac {(b d-a e)^3 (B d-A 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 = 0.12, size = 187, normalized size = 0.79 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (e x \left (12 a^3 B e^3+18 a^2 b e^2 (2 A e-2 B d+B e x)+6 a b^2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )+12 (b d-a e)^3 (B d-A e) \log (d+e x)\right )}{12 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.05, 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} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 260, normalized size = 1.10 \begin {gather*} \frac {3 \, B b^{3} e^{4} x^{4} - 4 \, {\left (B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 12 \, {\left (B b^{3} d^{3} e - {\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} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\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}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 428, normalized size = 1.81 \begin {gather*} {\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 )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B b^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, B b^{3} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b^{3} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) - 12 \, B b^{3} d^{3} x \mathrm {sgn}\left (b x + a\right ) + 12 \, B a b^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, A b^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 18 \, B a b^{2} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, A b^{3} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 36 \, B a b^{2} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + 12 \, A b^{3} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + 18 \, B a^{2} b x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 36 \, B a^{2} b d x e^{2} \mathrm {sgn}\left (b x + a\right ) - 36 \, A a b^{2} d x e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, A a^{2} b x e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 358, normalized size = 1.52 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (3 B \,b^{3} e^{4} x^{4}+4 A \,b^{3} e^{4} x^{3}+12 B a \,b^{2} e^{4} x^{3}-4 B \,b^{3} d \,e^{3} x^{3}+18 A a \,b^{2} e^{4} x^{2}-6 A \,b^{3} d \,e^{3} x^{2}+18 B \,a^{2} b \,e^{4} x^{2}-18 B a \,b^{2} d \,e^{3} x^{2}+6 B \,b^{3} d^{2} e^{2} x^{2}+12 A \,a^{3} e^{4} \ln \left (e x +d \right )-36 A \,a^{2} b d \,e^{3} \ln \left (e x +d \right )+36 A \,a^{2} b \,e^{4} x +36 A a \,b^{2} d^{2} e^{2} \ln \left (e x +d \right )-36 A a \,b^{2} d \,e^{3} x -12 A \,b^{3} d^{3} e \ln \left (e x +d \right )+12 A \,b^{3} d^{2} e^{2} x -12 B \,a^{3} d \,e^{3} \ln \left (e x +d \right )+12 B \,a^{3} e^{4} x +36 B \,a^{2} b \,d^{2} e^{2} \ln \left (e x +d \right )-36 B \,a^{2} b d \,e^{3} x -36 B a \,b^{2} d^{3} e \ln \left (e x +d \right )+36 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{4} \ln \left (e x +d \right )-12 B \,b^{3} d^{3} e x \right )}{12 \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 [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}}{d+e\,x} \,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}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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