3.1425 \(\int \frac {(a+b x)^4}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ -\frac {8 b^3 (c+d x)^{5/2} (b c-a d)}{5 d^5}+\frac {4 b^2 (c+d x)^{3/2} (b c-a d)^2}{d^5}-\frac {8 b \sqrt {c+d x} (b c-a d)^3}{d^5}-\frac {2 (b c-a d)^4}{d^5 \sqrt {c+d x}}+\frac {2 b^4 (c+d x)^{7/2}}{7 d^5} \]

[Out]

4*b^2*(-a*d+b*c)^2*(d*x+c)^(3/2)/d^5-8/5*b^3*(-a*d+b*c)*(d*x+c)^(5/2)/d^5+2/7*b^4*(d*x+c)^(7/2)/d^5-2*(-a*d+b*
c)^4/d^5/(d*x+c)^(1/2)-8*b*(-a*d+b*c)^3*(d*x+c)^(1/2)/d^5

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Rubi [A]  time = 0.04, antiderivative size = 123, 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 = {43} \[ -\frac {8 b^3 (c+d x)^{5/2} (b c-a d)}{5 d^5}+\frac {4 b^2 (c+d x)^{3/2} (b c-a d)^2}{d^5}-\frac {8 b \sqrt {c+d x} (b c-a d)^3}{d^5}-\frac {2 (b c-a d)^4}{d^5 \sqrt {c+d x}}+\frac {2 b^4 (c+d x)^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x)^(3/2),x]

[Out]

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

Rule 43

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)^4}{(c+d x)^{3/2}} \, dx &=\int \left (\frac {(-b c+a d)^4}{d^4 (c+d x)^{3/2}}-\frac {4 b (b c-a d)^3}{d^4 \sqrt {c+d x}}+\frac {6 b^2 (b c-a d)^2 \sqrt {c+d x}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{3/2}}{d^4}+\frac {b^4 (c+d x)^{5/2}}{d^4}\right ) \, dx\\ &=-\frac {2 (b c-a d)^4}{d^5 \sqrt {c+d x}}-\frac {8 b (b c-a d)^3 \sqrt {c+d x}}{d^5}+\frac {4 b^2 (b c-a d)^2 (c+d x)^{3/2}}{d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{5/2}}{5 d^5}+\frac {2 b^4 (c+d x)^{7/2}}{7 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 101, normalized size = 0.82 \[ \frac {2 \left (-28 b^3 (c+d x)^3 (b c-a d)+70 b^2 (c+d x)^2 (b c-a d)^2-140 b (c+d x) (b c-a d)^3-35 (b c-a d)^4+5 b^4 (c+d x)^4\right )}{35 d^5 \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x)^(3/2),x]

[Out]

(2*(-35*(b*c - a*d)^4 - 140*b*(b*c - a*d)^3*(c + d*x) + 70*b^2*(b*c - a*d)^2*(c + d*x)^2 - 28*b^3*(b*c - a*d)*
(c + d*x)^3 + 5*b^4*(c + d*x)^4))/(35*d^5*Sqrt[c + d*x])

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fricas [A]  time = 0.42, size = 192, normalized size = 1.56 \[ \frac {2 \, {\left (5 \, b^{4} d^{4} x^{4} - 128 \, b^{4} c^{4} + 448 \, a b^{3} c^{3} d - 560 \, a^{2} b^{2} c^{2} d^{2} + 280 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4} - 4 \, {\left (2 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{3} + 2 \, {\left (8 \, b^{4} c^{2} d^{2} - 28 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} c^{3} d - 56 \, a b^{3} c^{2} d^{2} + 70 \, a^{2} b^{2} c d^{3} - 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{35 \, {\left (d^{6} x + c d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b^4*d^4*x^4 - 128*b^4*c^4 + 448*a*b^3*c^3*d - 560*a^2*b^2*c^2*d^2 + 280*a^3*b*c*d^3 - 35*a^4*d^4 - 4*(
2*b^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 2*(8*b^4*c^2*d^2 - 28*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^2 - 4*(16*b^4*c^3*d - 5
6*a*b^3*c^2*d^2 + 70*a^2*b^2*c*d^3 - 35*a^3*b*d^4)*x)*sqrt(d*x + c)/(d^6*x + c*d^5)

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giac [B]  time = 1.08, size = 240, normalized size = 1.95 \[ -\frac {2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}}{\sqrt {d x + c} d^{5}} + \frac {2 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} d^{30} - 28 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c d^{30} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{2} d^{30} - 140 \, \sqrt {d x + c} b^{4} c^{3} d^{30} + 28 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} d^{31} - 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c d^{31} + 420 \, \sqrt {d x + c} a b^{3} c^{2} d^{31} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} d^{32} - 420 \, \sqrt {d x + c} a^{2} b^{2} c d^{32} + 140 \, \sqrt {d x + c} a^{3} b d^{33}\right )}}{35 \, d^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(sqrt(d*x + c)*d^5) + 2/35*(5*(d*x
+ c)^(7/2)*b^4*d^30 - 28*(d*x + c)^(5/2)*b^4*c*d^30 + 70*(d*x + c)^(3/2)*b^4*c^2*d^30 - 140*sqrt(d*x + c)*b^4*
c^3*d^30 + 28*(d*x + c)^(5/2)*a*b^3*d^31 - 140*(d*x + c)^(3/2)*a*b^3*c*d^31 + 420*sqrt(d*x + c)*a*b^3*c^2*d^31
+ 70*(d*x + c)^(3/2)*a^2*b^2*d^32 - 420*sqrt(d*x + c)*a^2*b^2*c*d^32 + 140*sqrt(d*x + c)*a^3*b*d^33)/d^35

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maple [A]  time = 0.01, size = 186, normalized size = 1.51 \[ -\frac {2 \left (-5 b^{4} x^{4} d^{4}-28 a \,b^{3} d^{4} x^{3}+8 b^{4} c \,d^{3} x^{3}-70 a^{2} b^{2} d^{4} x^{2}+56 a \,b^{3} c \,d^{3} x^{2}-16 b^{4} c^{2} d^{2} x^{2}-140 a^{3} b \,d^{4} x +280 a^{2} b^{2} c \,d^{3} x -224 a \,b^{3} c^{2} d^{2} x +64 b^{4} c^{3} d x +35 a^{4} d^{4}-280 a^{3} b c \,d^{3}+560 a^{2} b^{2} c^{2} d^{2}-448 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{35 \sqrt {d x +c}\, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c)^(3/2),x)

[Out]

-2/35/(d*x+c)^(1/2)*(-5*b^4*d^4*x^4-28*a*b^3*d^4*x^3+8*b^4*c*d^3*x^3-70*a^2*b^2*d^4*x^2+56*a*b^3*c*d^3*x^2-16*
b^4*c^2*d^2*x^2-140*a^3*b*d^4*x+280*a^2*b^2*c*d^3*x-224*a*b^3*c^2*d^2*x+64*b^4*c^3*d*x+35*a^4*d^4-280*a^3*b*c*
d^3+560*a^2*b^2*c^2*d^2-448*a*b^3*c^3*d+128*b^4*c^4)/d^5

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maxima [A]  time = 1.35, size = 189, normalized size = 1.54 \[ \frac {2 \, {\left (\frac {5 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} - 28 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 70 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 140 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {d x + c}}{d^{4}} - \frac {35 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}}{\sqrt {d x + c} d^{4}}\right )}}{35 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/35*((5*(d*x + c)^(7/2)*b^4 - 28*(b^4*c - a*b^3*d)*(d*x + c)^(5/2) + 70*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)
*(d*x + c)^(3/2) - 140*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(d*x + c))/d^4 - 35*(b^4*c^
4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(sqrt(d*x + c)*d^4))/d

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mupad [B]  time = 0.06, size = 153, normalized size = 1.24 \[ \frac {2\,b^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}-\frac {2\,a^4\,d^4-8\,a^3\,b\,c\,d^3+12\,a^2\,b^2\,c^2\,d^2-8\,a\,b^3\,c^3\,d+2\,b^4\,c^4}{d^5\,\sqrt {c+d\,x}}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(c + d*x)^(3/2),x)

[Out]

(2*b^4*(c + d*x)^(7/2))/(7*d^5) - ((8*b^4*c - 8*a*b^3*d)*(c + d*x)^(5/2))/(5*d^5) - (2*a^4*d^4 + 2*b^4*c^4 + 1
2*a^2*b^2*c^2*d^2 - 8*a*b^3*c^3*d - 8*a^3*b*c*d^3)/(d^5*(c + d*x)^(1/2)) + (4*b^2*(a*d - b*c)^2*(c + d*x)^(3/2
))/d^5 + (8*b*(a*d - b*c)^3*(c + d*x)^(1/2))/d^5

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sympy [A]  time = 32.87, size = 168, normalized size = 1.37 \[ \frac {2 b^{4} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (8 a b^{3} d - 8 b^{4} c\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (12 a^{2} b^{2} d^{2} - 24 a b^{3} c d + 12 b^{4} c^{2}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (8 a^{3} b d^{3} - 24 a^{2} b^{2} c d^{2} + 24 a b^{3} c^{2} d - 8 b^{4} c^{3}\right )}{d^{5}} - \frac {2 \left (a d - b c\right )^{4}}{d^{5} \sqrt {c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(d*x+c)**(3/2),x)

[Out]

2*b**4*(c + d*x)**(7/2)/(7*d**5) + (c + d*x)**(5/2)*(8*a*b**3*d - 8*b**4*c)/(5*d**5) + (c + d*x)**(3/2)*(12*a*
*2*b**2*d**2 - 24*a*b**3*c*d + 12*b**4*c**2)/(3*d**5) + sqrt(c + d*x)*(8*a**3*b*d**3 - 24*a**2*b**2*c*d**2 + 2
4*a*b**3*c**2*d - 8*b**4*c**3)/d**5 - 2*(a*d - b*c)**4/(d**5*sqrt(c + d*x))

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