3.1480 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(5/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (4*d*(c + d*x)^(5/2))/(33*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(5/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (32*d^3*(c + d*x)^(5/2))/(1155*(b*c - a*d
)^4*(a + b*x)^(5/2))

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Rubi [A]  time = 0.0283103, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac{4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (4*d*(c + d*x)^(5/2))/(33*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(5/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (32*d^3*(c + d*x)^(5/2))/(1155*(b*c - a*d
)^4*(a + b*x)^(5/2))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx &=-\frac{2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac{(6 d) \int \frac{(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac{2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}+\frac{\left (8 d^2\right ) \int \frac{(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac{16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac{\left (16 d^3\right ) \int \frac{(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac{2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac{16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac{32 d^3 (c+d x)^{5/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0634928, size = 118, normalized size = 0.87 \[ \frac{2 (c+d x)^{5/2} \left (99 a^2 b d^2 (2 d x-5 c)+231 a^3 d^3+11 a b^2 d \left (35 c^2-20 c d x+8 d^2 x^2\right )+b^3 \left (70 c^2 d x-105 c^3-40 c d^2 x^2+16 d^3 x^3\right )\right )}{1155 (a+b x)^{11/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(2*(c + d*x)^(5/2)*(231*a^3*d^3 + 99*a^2*b*d^2*(-5*c + 2*d*x) + 11*a*b^2*d*(35*c^2 - 20*c*d*x + 8*d^2*x^2) + b
^3*(-105*c^3 + 70*c^2*d*x - 40*c*d^2*x^2 + 16*d^3*x^3)))/(1155*(b*c - a*d)^4*(a + b*x)^(11/2))

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Maple [A]  time = 0.009, size = 171, normalized size = 1.3 \begin{align*}{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+176\,a{b}^{2}{d}^{3}{x}^{2}-80\,{b}^{3}c{d}^{2}{x}^{2}+396\,{a}^{2}b{d}^{3}x-440\,a{b}^{2}c{d}^{2}x+140\,{b}^{3}{c}^{2}dx+462\,{a}^{3}{d}^{3}-990\,{a}^{2}bc{d}^{2}+770\,a{b}^{2}{c}^{2}d-210\,{b}^{3}{c}^{3}}{1155\,{d}^{4}{a}^{4}-4620\,b{d}^{3}c{a}^{3}+6930\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-4620\,{b}^{3}d{c}^{3}a+1155\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(b*x+a)^(13/2),x)

[Out]

2/1155*(d*x+c)^(5/2)*(16*b^3*d^3*x^3+88*a*b^2*d^3*x^2-40*b^3*c*d^2*x^2+198*a^2*b*d^3*x-220*a*b^2*c*d^2*x+70*b^
3*c^2*d*x+231*a^3*d^3-495*a^2*b*c*d^2+385*a*b^2*c^2*d-105*b^3*c^3)/(b*x+a)^(11/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2
*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 100.004, size = 1328, normalized size = 9.76 \begin{align*} \frac{2 \,{\left (16 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 385 \, a b^{2} c^{4} d - 495 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} - 8 \,{\left (b^{3} c d^{4} - 11 \, a b^{2} d^{5}\right )} x^{4} + 2 \,{\left (3 \, b^{3} c^{2} d^{3} - 22 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} -{\left (5 \, b^{3} c^{3} d^{2} - 33 \, a b^{2} c^{2} d^{3} + 99 \, a^{2} b c d^{4} - 231 \, a^{3} d^{5}\right )} x^{2} - 2 \,{\left (70 \, b^{3} c^{4} d - 275 \, a b^{2} c^{3} d^{2} + 396 \, a^{2} b c^{2} d^{3} - 231 \, a^{3} c d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1155 \,{\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} +{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \,{\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \,{\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \,{\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \,{\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*b^3*d^5*x^5 - 105*b^3*c^5 + 385*a*b^2*c^4*d - 495*a^2*b*c^3*d^2 + 231*a^3*c^2*d^3 - 8*(b^3*c*d^4 -
11*a*b^2*d^5)*x^4 + 2*(3*b^3*c^2*d^3 - 22*a*b^2*c*d^4 + 99*a^2*b*d^5)*x^3 - (5*b^3*c^3*d^2 - 33*a*b^2*c^2*d^3
+ 99*a^2*b*c*d^4 - 231*a^3*d^5)*x^2 - 2*(70*b^3*c^4*d - 275*a*b^2*c^3*d^2 + 396*a^2*b*c^2*d^3 - 231*a^3*c*d^4)
*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^4*c^4 - 4*a^7*b^3*c^3*d + 6*a^8*b^2*c^2*d^2 - 4*a^9*b*c*d^3 + a^10*d^4
+ (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 - 4*a^2*b^
8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b
^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^4 + 20*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^
6*b^4*c*d^3 + a^7*b^3*d^4)*x^3 + 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8
*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 2.09559, size = 2461, normalized size = 18.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="giac")

[Out]

64/1155*(sqrt(b*d)*b^14*c^7*d^5*abs(b) - 7*sqrt(b*d)*a*b^13*c^6*d^6*abs(b) + 21*sqrt(b*d)*a^2*b^12*c^5*d^7*abs
(b) - 35*sqrt(b*d)*a^3*b^11*c^4*d^8*abs(b) + 35*sqrt(b*d)*a^4*b^10*c^3*d^9*abs(b) - 21*sqrt(b*d)*a^5*b^9*c^2*d
^10*abs(b) + 7*sqrt(b*d)*a^6*b^8*c*d^11*abs(b) - sqrt(b*d)*a^7*b^7*d^12*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^12*c^6*d^5*abs(b) + 66*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^11*c^5*d^6*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^10*c^4*d^7*abs(b) + 220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^9*c^3*d^8*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^4*b^8*c^2*d^9*abs(b) + 66*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*a^5*b^7*c*d^10*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^6*b^6*d^11*abs(b) + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*b^10*c^5*d^5*abs(b) - 275*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^
9*c^4*d^6*abs(b) + 550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^8*c^3
*d^7*abs(b) - 550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^7*c^2*d^8*
abs(b) + 275*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^6*c*d^9*abs(b)
- 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^5*d^10*abs(b) - 165*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^8*c^4*d^5*abs(b) + 660*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^7*c^3*d^6*abs(b) - 990*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^6*c^2*d^7*abs(b) + 660*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^5*c*d^8*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^4*d^9*abs(b) - 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^6*c^3*d^5*abs(b) + 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*a*b^5*c^2*d^6*abs(b) - 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^2*b^4*c*d^7*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^8*a^3*b^3*d^8*abs(b) - 2541*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^10*b^4*c^2*d^5*abs(b) + 5082*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10
*a*b^3*c*d^6*abs(b) - 2541*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^
2*d^7*abs(b) - 2079*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^2*c*d^5*abs
(b) + 2079*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b*d^6*abs(b) - 1155*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*d^5*abs(b))/(b^2*c - a*b*d - (sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^11