\(\int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx\) [1415]

Optimal result

Integrand size = 17, antiderivative size = 96 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=-\frac {2 (b c-a d)^3 \sqrt {c+d x}}{d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{5/2}}{5 d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4} \]

[Out]

2*b*(-a*d+b*c)^2*(d*x+c)^(3/2)/d^4-6/5*b^2*(-a*d+b*c)*(d*x+c)^(5/2)/d^4+2/7*b^3*(d*x+c)^(7/2)/d^4-2*(-a*d+b*c)
^3*(d*x+c)^(1/2)/d^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=-\frac {6 b^2 (c+d x)^{5/2} (b c-a d)}{5 d^4}+\frac {2 b (c+d x)^{3/2} (b c-a d)^2}{d^4}-\frac {2 \sqrt {c+d x} (b c-a d)^3}{d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4} \]

[In]

Int[(a + b*x)^3/Sqrt[c + d*x],x]

[Out]

(-2*(b*c - a*d)^3*Sqrt[c + d*x])/d^4 + (2*b*(b*c - a*d)^2*(c + d*x)^(3/2))/d^4 - (6*b^2*(b*c - a*d)*(c + d*x)^
(5/2))/(5*d^4) + (2*b^3*(c + d*x)^(7/2))/(7*d^4)

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3}{d^3 \sqrt {c+d x}}+\frac {3 b (b c-a d)^2 \sqrt {c+d x}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{3/2}}{d^3}+\frac {b^3 (c+d x)^{5/2}}{d^3}\right ) \, dx \\ & = -\frac {2 (b c-a d)^3 \sqrt {c+d x}}{d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{5/2}}{5 d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (35 a^3 d^3+35 a^2 b d^2 (-2 c+d x)+7 a b^2 d \left (8 c^2-4 c d x+3 d^2 x^2\right )+b^3 \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )}{35 d^4} \]

[In]

Integrate[(a + b*x)^3/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x]*(35*a^3*d^3 + 35*a^2*b*d^2*(-2*c + d*x) + 7*a*b^2*d*(8*c^2 - 4*c*d*x + 3*d^2*x^2) + b^3*(-16*
c^3 + 8*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3)))/(35*d^4)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {3}{2}}+2 \left (a d -b c \right )^{3} \sqrt {d x +c}}{d^{4}}\)	\(76\)
default	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {3}{2}}+2 \left (a d -b c \right )^{3} \sqrt {d x +c}}{d^{4}}\)	\(76\)
pseudoelliptic	\(\frac {2 \sqrt {d x +c}\, \left (\left (\frac {1}{7} b^{3} x^{3}+\frac {3}{5} a \,b^{2} x^{2}+a^{2} b x +a^{3}\right ) d^{3}-2 b \left (\frac {3}{35} b^{2} x^{2}+\frac {2}{5} a b x +a^{2}\right ) c \,d^{2}+\frac {8 \left (\frac {b x}{7}+a \right ) b^{2} c^{2} d}{5}-\frac {16 b^{3} c^{3}}{35}\right )}{d^{4}}\)	\(92\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (5 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}-6 x^{2} b^{3} c \,d^{2}+35 x \,a^{2} b \,d^{3}-28 x a \,b^{2} c \,d^{2}+8 x \,b^{3} c^{2} d +35 a^{3} d^{3}-70 a^{2} b c \,d^{2}+56 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{35 d^{4}}\)	\(116\)
trager	\(\frac {2 \sqrt {d x +c}\, \left (5 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}-6 x^{2} b^{3} c \,d^{2}+35 x \,a^{2} b \,d^{3}-28 x a \,b^{2} c \,d^{2}+8 x \,b^{3} c^{2} d +35 a^{3} d^{3}-70 a^{2} b c \,d^{2}+56 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{35 d^{4}}\)	\(116\)
risch	\(\frac {2 \sqrt {d x +c}\, \left (5 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}-6 x^{2} b^{3} c \,d^{2}+35 x \,a^{2} b \,d^{3}-28 x a \,b^{2} c \,d^{2}+8 x \,b^{3} c^{2} d +35 a^{3} d^{3}-70 a^{2} b c \,d^{2}+56 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{35 d^{4}}\)	\(116\)

[In]

int((b*x+a)^3/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (5 \, b^{3} d^{3} x^{3} - 16 \, b^{3} c^{3} + 56 \, a b^{2} c^{2} d - 70 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 3 \, {\left (2 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + {\left (8 \, b^{3} c^{2} d - 28 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{35 \, d^{4}} \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*b^3*d^3*x^3 - 16*b^3*c^3 + 56*a*b^2*c^2*d - 70*a^2*b*c*d^2 + 35*a^3*d^3 - 3*(2*b^3*c*d^2 - 7*a*b^2*d^3
)*x^2 + (8*b^3*c^2*d - 28*a*b^2*c*d^2 + 35*a^2*b*d^3)*x)*sqrt(d*x + c)/d^4

Sympy [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{3}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (3 a b^{2} d - 3 b^{3} c\right )}{5 d^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} b d^{2} - 6 a b^{2} c d + 3 b^{3} c^{2}\right )}{3 d^{3}} + \frac {\sqrt {c + d x} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{d^{3}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x+a)**3/(d*x+c)**(1/2),x)

[Out]

Piecewise((2*(b**3*(c + d*x)**(7/2)/(7*d**3) + (c + d*x)**(5/2)*(3*a*b**2*d - 3*b**3*c)/(5*d**3) + (c + d*x)**
(3/2)*(3*a**2*b*d**2 - 6*a*b**2*c*d + 3*b**3*c**2)/(3*d**3) + sqrt(c + d*x)*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a
*b**2*c**2*d - b**3*c**3)/d**3)/d, Ne(d, 0)), (Piecewise((a**3*x, Eq(b, 0)), ((a + b*x)**4/(4*b), True))/sqrt(
c), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (35 \, \sqrt {d x + c} a^{3} + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac {{\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (35 \, \sqrt {d x + c} a^{3} + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac {{\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx=\frac {2\,b^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{d^4} \]

[In]

int((a + b*x)^3/(c + d*x)^(1/2),x)

[Out]

(2*b^3*(c + d*x)^(7/2))/(7*d^4) - ((6*b^3*c - 6*a*b^2*d)*(c + d*x)^(5/2))/(5*d^4) + (2*(a*d - b*c)^3*(c + d*x)
^(1/2))/d^4 + (2*b*(a*d - b*c)^2*(c + d*x)^(3/2))/d^4