3.15.26 \(\int \frac {(a+b x)^3}{(c+d x)^{3/2}} \, dx\) [1426]

Optimal. Leaf size=94 \[ \frac {2 (b c-a d)^3}{d^4 \sqrt {c+d x}}+\frac {6 b (b c-a d)^2 \sqrt {c+d x}}{d^4}-\frac {2 b^2 (b c-a d) (c+d x)^{3/2}}{d^4}+\frac {2 b^3 (c+d x)^{5/2}}{5 d^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} -\frac {2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac {6 b \sqrt {c+d x} (b c-a d)^2}{d^4}+\frac {2 (b c-a d)^3}{d^4 \sqrt {c+d x}}+\frac {2 b^3 (c+d x)^{5/2}}{5 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^(3/2),x]

[Out]

(2*(b*c - a*d)^3)/(d^4*Sqrt[c + d*x]) + (6*b*(b*c - a*d)^2*Sqrt[c + d*x])/d^4 - (2*b^2*(b*c - a*d)*(c + d*x)^(
3/2))/d^4 + (2*b^3*(c + d*x)^(5/2))/(5*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*} \int \frac {(a+b x)^3}{(c+d x)^{3/2}} \, dx &=\int \left (\frac {(-b c+a d)^3}{d^3 (c+d x)^{3/2}}+\frac {3 b (b c-a d)^2}{d^3 \sqrt {c+d x}}-\frac {3 b^2 (b c-a d) \sqrt {c+d x}}{d^3}+\frac {b^3 (c+d x)^{3/2}}{d^3}\right ) \, dx\\ &=\frac {2 (b c-a d)^3}{d^4 \sqrt {c+d x}}+\frac {6 b (b c-a d)^2 \sqrt {c+d x}}{d^4}-\frac {2 b^2 (b c-a d) (c+d x)^{3/2}}{d^4}+\frac {2 b^3 (c+d x)^{5/2}}{5 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 99, normalized size = 1.05 \begin {gather*} \frac {2 \left (-5 a^3 d^3+15 a^2 b d^2 (2 c+d x)+5 a b^2 d \left (-8 c^2-4 c d x+d^2 x^2\right )+b^3 \left (16 c^3+8 c^2 d x-2 c d^2 x^2+d^3 x^3\right )\right )}{5 d^4 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^(3/2),x]

[Out]

(2*(-5*a^3*d^3 + 15*a^2*b*d^2*(2*c + d*x) + 5*a*b^2*d*(-8*c^2 - 4*c*d*x + d^2*x^2) + b^3*(16*c^3 + 8*c^2*d*x -
2*c*d^2*x^2 + d^3*x^3)))/(5*d^4*Sqrt[c + d*x])

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Maple [A]
time = 0.19, size = 136, normalized size = 1.45

method	result	size
risch	\(\frac {2 b \left (b^{2} x^{2} d^{2}+5 a b \,d^{2} x -3 b^{2} c d x +15 a^{2} d^{2}-25 a b c d +11 b^{2} c^{2}\right ) \sqrt {d x +c}}{5 d^{4}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{4} \sqrt {d x +c}}\)	\(112\)
gosper	\(-\frac {2 \left (-b^{3} x^{3} d^{3}-5 d^{3} a \,x^{2} b^{2}+2 b^{3} c \,d^{2} x^{2}-15 a^{2} b \,d^{3} x +20 a \,b^{2} c \,d^{2} x -8 b^{3} c^{2} d x +5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{5 \sqrt {d x +c}\, d^{4}}\)	\(116\)
trager	\(-\frac {2 \left (-b^{3} x^{3} d^{3}-5 d^{3} a \,x^{2} b^{2}+2 b^{3} c \,d^{2} x^{2}-15 a^{2} b \,d^{3} x +20 a \,b^{2} c \,d^{2} x -8 b^{3} c^{2} d x +5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{5 \sqrt {d x +c}\, d^{4}}\)	\(116\)
derivativedivides	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {5}{2}}}{5}+2 a \,b^{2} d \left (d x +c \right )^{\frac {3}{2}}-2 b^{3} c \left (d x +c \right )^{\frac {3}{2}}+6 a^{2} b \,d^{2} \sqrt {d x +c}-12 a \,b^{2} c d \sqrt {d x +c}+6 b^{3} c^{2} \sqrt {d x +c}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{\sqrt {d x +c}}}{d^{4}}\)	\(136\)
default	\(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {5}{2}}}{5}+2 a \,b^{2} d \left (d x +c \right )^{\frac {3}{2}}-2 b^{3} c \left (d x +c \right )^{\frac {3}{2}}+6 a^{2} b \,d^{2} \sqrt {d x +c}-12 a \,b^{2} c d \sqrt {d x +c}+6 b^{3} c^{2} \sqrt {d x +c}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{\sqrt {d x +c}}}{d^{4}}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 125, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (\frac {{\left (d x + c\right )}^{\frac {5}{2}} b^{3} - 5 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 15 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {d x + c}}{d^{3}} + \frac {5 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt {d x + c} d^{3}}\right )}}{5 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.21, size = 124, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (b^{3} d^{3} x^{3} + 16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} - {\left (2 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} + {\left (8 \, b^{3} c^{2} d - 20 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{5 \, {\left (d^{5} x + c d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b^3*d^3*x^3 + 16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3 - (2*b^3*c*d^2 - 5*a*b^2*d^3)*x^2
+ (8*b^3*c^2*d - 20*a*b^2*c*d^2 + 15*a^2*b*d^3)*x)*sqrt(d*x + c)/(d^5*x + c*d^4)

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Sympy [A]
time = 8.90, size = 109, normalized size = 1.16 \begin {gather*} \frac {2 b^{3} \left (c + d x\right )^{\frac {5}{2}}}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (6 a b^{2} d - 6 b^{3} c\right )}{3 d^{4}} + \frac {\sqrt {c + d x} \left (6 a^{2} b d^{2} - 12 a b^{2} c d + 6 b^{3} c^{2}\right )}{d^{4}} - \frac {2 \left (a d - b c\right )^{3}}{d^{4} \sqrt {c + d x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*(c + d*x)**(5/2)/(5*d**4) + (c + d*x)**(3/2)*(6*a*b**2*d - 6*b**3*c)/(3*d**4) + sqrt(c + d*x)*(6*a**2*b
*d**2 - 12*a*b**2*c*d + 6*b**3*c**2)/d**4 - 2*(a*d - b*c)**3/(d**4*sqrt(c + d*x))

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Giac [A]
time = 1.15, size = 152, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt {d x + c} d^{4}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {5}{2}} b^{3} d^{16} - 5 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c d^{16} + 15 \, \sqrt {d x + c} b^{3} c^{2} d^{16} + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} d^{17} - 30 \, \sqrt {d x + c} a b^{2} c d^{17} + 15 \, \sqrt {d x + c} a^{2} b d^{18}\right )}}{5 \, d^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(sqrt(d*x + c)*d^4) + 2/5*((d*x + c)^(5/2)*b^3*d^16 - 5*
(d*x + c)^(3/2)*b^3*c*d^16 + 15*sqrt(d*x + c)*b^3*c^2*d^16 + 5*(d*x + c)^(3/2)*a*b^2*d^17 - 30*sqrt(d*x + c)*a
*b^2*c*d^17 + 15*sqrt(d*x + c)*a^2*b*d^18)/d^20

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Mupad [B]
time = 0.08, size = 114, normalized size = 1.21 \begin {gather*} \frac {2\,b^3\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^4}-\frac {2\,a^3\,d^3-6\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-2\,b^3\,c^3}{d^4\,\sqrt {c+d\,x}}+\frac {6\,b\,{\left (a\,d-b\,c\right )}^2\,\sqrt {c+d\,x}}{d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/(c + d*x)^(3/2),x)

[Out]

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

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