3.4.40 \(\int \sqrt {d+e x} (b x+c x^2) \, dx\) [340]

Optimal. Leaf size=68 \[ \frac {2 d (c d-b e) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \]

[Out]

2/3*d*(-b*e+c*d)*(e*x+d)^(3/2)/e^3-2/5*(-b*e+2*c*d)*(e*x+d)^(5/2)/e^3+2/7*c*(e*x+d)^(7/2)/e^3

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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} -\frac {2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac {2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^3) + (2*c*(d + e*x)^(7/2))/
(7*e^3)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) \sqrt {d+e x}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{3/2}}{e^2}+\frac {c (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 d (c d-b e) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.74 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (7 b e (-2 d+3 e x)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(3/2)*(7*b*e*(-2*d + 3*e*x) + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)

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Maple [A]
time = 0.41, size = 52, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-15 c \,x^{2} e^{2}-21 b \,e^{2} x +12 c d e x +14 b d e -8 c \,d^{2}\right )}{105 e^{3}}\)	\(47\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(52\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(52\)
trager	\(-\frac {2 \left (-15 x^{3} e^{3} c -21 e^{3} b \,x^{2}-3 c d \,e^{2} x^{2}-7 b d \,e^{2} x +4 c \,d^{2} e x +14 b \,d^{2} e -8 d^{3} c \right ) \sqrt {e x +d}}{105 e^{3}}\)	\(71\)
risch	\(-\frac {2 \left (-15 x^{3} e^{3} c -21 e^{3} b \,x^{2}-3 c d \,e^{2} x^{2}-7 b d \,e^{2} x +4 c \,d^{2} e x +14 b \,d^{2} e -8 d^{3} c \right ) \sqrt {e x +d}}{105 e^{3}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

2/e^3*(1/7*c*(e*x+d)^(7/2)+1/5*(b*e-2*c*d)*(e*x+d)^(5/2)-1/3*d*(b*e-c*d)*(e*x+d)^(3/2))

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Maxima [A]
time = 0.29, size = 58, normalized size = 0.85 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c - 21 \, {\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (c d^{2} - b d e\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/105*(15*(x*e + d)^(7/2)*c - 21*(2*c*d - b*e)*(x*e + d)^(5/2) + 35*(c*d^2 - b*d*e)*(x*e + d)^(3/2))*e^(-3)

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Fricas [A]
time = 1.71, size = 70, normalized size = 1.03 \begin {gather*} \frac {2}{105} \, {\left (8 \, c d^{3} + 3 \, {\left (5 \, c x^{3} + 7 \, b x^{2}\right )} e^{3} + {\left (3 \, c d x^{2} + 7 \, b d x\right )} e^{2} - 2 \, {\left (2 \, c d^{2} x + 7 \, b d^{2}\right )} e\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/105*(8*c*d^3 + 3*(5*c*x^3 + 7*b*x^2)*e^3 + (3*c*d*x^2 + 7*b*d*x)*e^2 - 2*(2*c*d^2*x + 7*b*d^2)*e)*sqrt(x*e +
d)*e^(-3)

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Sympy [A]
time = 1.18, size = 66, normalized size = 0.97 \begin {gather*} \frac {2 \left (\frac {c \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(c*x**2+b*x),x)

[Out]

2*(c*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(b*e - 2*c*d)/(5*e**2) + (d + e*x)**(3/2)*(-b*d*e + c*d**2)/
(3*e**2))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (58) = 116\).
time = 1.47, size = 165, normalized size = 2.43 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d e^{\left (-2\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b e^{\left (-1\right )} + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c e^{\left (-2\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*d*e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*c*d*e^(-2) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b*e^(-1) +
3*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c*e^(-2))*e^(-1)

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Mupad [B]
time = 0.21, size = 52, normalized size = 0.76 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,c\,{\left (d+e\,x\right )}^2+35\,c\,d^2+21\,b\,e\,\left (d+e\,x\right )-42\,c\,d\,\left (d+e\,x\right )-35\,b\,d\,e\right )}{105\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d + e*x)^(1/2),x)

[Out]

(2*(d + e*x)^(3/2)*(15*c*(d + e*x)^2 + 35*c*d^2 + 21*b*e*(d + e*x) - 42*c*d*(d + e*x) - 35*b*d*e))/(105*e^3)

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