\(\int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [2095]

Optimal result

Integrand size = 35, antiderivative size = 152 \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)} \]

[Out]

2/3*(-a*e+b*d)^2*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-4/5*b*(-a*e+b*d)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/
e^3/(b*x+a)+2/7*b^2*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45} \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^3 (a+b x)} \]

[In]

Int[(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b*x)) - (4*b*(b*d - a*e)*(d + e*x)
^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) + (2*b^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(7*e^3*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) \sqrt {d+e x} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 \sqrt {d+e x} \, 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)^2 \sqrt {d+e x}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{3/2}}{e^2}+\frac {b^2 (d+e x)^{5/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (35 a^2 e^2+14 a b e (-2 d+3 e x)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3 (a+b x)} \]

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^
2)))/(105*e^3*(a + b*x))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45

method	result	size
default	\(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {3}{2}} \left (15 b^{2} e^{2} x^{2}+42 a b \,e^{2} x -12 b^{2} d e x +35 e^{2} a^{2}-28 a b d e +8 b^{2} d^{2}\right )}{105 e^{3}}\)	\(69\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 b^{2} e^{2} x^{2}+42 a b \,e^{2} x -12 b^{2} d e x +35 e^{2} a^{2}-28 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{105 e^{3} \left (b x +a \right )}\)	\(79\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (15 b^{2} x^{3} e^{3}+42 a b \,e^{3} x^{2}+3 b^{2} d \,e^{2} x^{2}+35 a^{2} e^{3} x +14 a b d \,e^{2} x -4 b^{2} d^{2} e x +35 a^{2} d \,e^{2}-28 a b \,d^{2} e +8 b^{2} d^{3}\right ) \sqrt {e x +d}}{105 \left (b x +a \right ) e^{3}}\)	\(116\)

[In]

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

[Out]

2/105*csgn(b*x+a)*(e*x+d)^(3/2)*(15*b^2*e^2*x^2+42*a*b*e^2*x-12*b^2*d*e*x+35*a^2*e^2-28*a*b*d*e+8*b^2*d^2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (15 \, b^{2} e^{3} x^{3} + 8 \, b^{2} d^{3} - 28 \, a b d^{2} e + 35 \, a^{2} d e^{2} + 3 \, {\left (b^{2} d e^{2} + 14 \, a b e^{3}\right )} x^{2} - {\left (4 \, b^{2} d^{2} e - 14 \, a b d e^{2} - 35 \, a^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]

[In]

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

[Out]

2/105*(15*b^2*e^3*x^3 + 8*b^2*d^3 - 28*a*b*d^2*e + 35*a^2*d*e^2 + 3*(b^2*d*e^2 + 14*a*b*e^3)*x^2 - (4*b^2*d^2*
e - 14*a*b*d*e^2 - 35*a^2*e^3)*x)*sqrt(e*x + d)/e^3

Sympy [F]

\[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \left (a + b x\right ) \sqrt {d + e x} \sqrt {\left (a + b x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*x)*sqrt(d + e*x)*sqrt((a + b*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.79 \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e + {\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt {e x + d} a}{15 \, e^{2}} + \frac {2 \, {\left (15 \, b e^{3} x^{3} + 8 \, b d^{3} - 14 \, a d^{2} e + 3 \, {\left (b d e^{2} + 7 \, a e^{3}\right )} x^{2} - {\left (4 \, b d^{2} e - 7 \, a d e^{2}\right )} x\right )} \sqrt {e x + d} b}{105 \, e^{3}} \]

[In]

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

[Out]

2/15*(3*b*e^2*x^2 - 2*b*d^2 + 5*a*d*e + (b*d*e + 5*a*e^2)*x)*sqrt(e*x + d)*a/e^2 + 2/105*(15*b*e^3*x^3 + 8*b*d
^3 - 14*a*d^2*e + 3*(b*d*e^2 + 7*a*e^3)*x^2 - (4*b*d^2*e - 7*a*d*e^2)*x)*sqrt(e*x + d)*b/e^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (107) = 214\).

Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.55 \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (105 \, \sqrt {e x + d} a^{2} d \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {70 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b d \mathrm {sgn}\left (b x + a\right )}{e} + \frac {7 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2} d \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {14 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a b \mathrm {sgn}\left (b x + a\right )}{e} + \frac {3 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{2} \mathrm {sgn}\left (b x + a\right )}{e^{2}}\right )}}{105 \, e} \]

[In]

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

[Out]

2/105*(105*sqrt(e*x + d)*a^2*d*sgn(b*x + a) + 35*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*sgn(b*x + a) + 70*(
(e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b*d*sgn(b*x + a)/e + 7*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*s
qrt(e*x + d)*d^2)*b^2*d*sgn(b*x + a)/e^2 + 14*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2
)*a*b*sgn(b*x + a)/e + 3*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)
*d^3)*b^2*sgn(b*x + a)/e^2)/e

Mupad [F(-1)]

Timed out. \[ \int (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )\,\sqrt {d+e\,x} \,d x \]

[In]

int(((a + b*x)^2)^(1/2)*(a + b*x)*(d + e*x)^(1/2),x)

[Out]

int(((a + b*x)^2)^(1/2)*(a + b*x)*(d + e*x)^(1/2), x)