3.608 \(\int \frac{(a+c x^2)^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=200 \[ \frac{2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7} \]

[Out]

(2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^7 - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c^2*(
5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (12*c^3*d*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(1
3*e^7)

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Rubi [A]  time = 0.0822133, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {697} \[ \frac{2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^7 - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c^2*(
5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (12*c^3*d*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(1
3*e^7)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^3}{\sqrt{d+e x}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 \sqrt{d+e x}}-\frac{6 c d \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{5/2}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^6}-\frac{6 c^3 d (d+e x)^{9/2}}{e^6}+\frac{c^3 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}{e^7}-\frac{4 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^7}+\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac{2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}\\ \end{align*}

Mathematica [A]  time = 0.117646, size = 171, normalized size = 0.86 \[ \frac{2 \sqrt{d+e x} \left (3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+15015 a^3 e^6+143 a c^2 e^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-512 d^5 e x+1024 d^6-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4 - 64*d
^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d
^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(15015*e^7)

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Maple [A]  time = 0.046, size = 205, normalized size = 1. \begin{align*}{\frac{2310\,{c}^{3}{x}^{6}{e}^{6}-2520\,{c}^{3}d{x}^{5}{e}^{5}+10010\,a{c}^{2}{e}^{6}{x}^{4}+2800\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-11440\,a{c}^{2}d{e}^{5}{x}^{3}-3200\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+18018\,{a}^{2}c{e}^{6}{x}^{2}+13728\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-24024\,{a}^{2}cd{e}^{5}x-18304\,a{c}^{2}{d}^{3}{e}^{3}x-5120\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}+48048\,{a}^{2}c{d}^{2}{e}^{4}+36608\,{d}^{4}{e}^{2}a{c}^{2}+10240\,{c}^{3}{d}^{6}}{15015\,{e}^{7}}\sqrt{ex+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^(1/2),x)

[Out]

2/15015*(e*x+d)^(1/2)*(1155*c^3*e^6*x^6-1260*c^3*d*e^5*x^5+5005*a*c^2*e^6*x^4+1400*c^3*d^2*e^4*x^4-5720*a*c^2*
d*e^5*x^3-1600*c^3*d^3*e^3*x^3+9009*a^2*c*e^6*x^2+6864*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2-12012*a^2*c*d*e^
5*x-9152*a*c^2*d^3*e^3*x-2560*c^3*d^5*e*x+15015*a^3*e^6+24024*a^2*c*d^2*e^4+18304*a*c^2*d^4*e^2+5120*c^3*d^6)/
e^7

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Maxima [A]  time = 1.47941, size = 286, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (15015 \, \sqrt{e x + d} a^{3} + \frac{3003 \,{\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^{2} c}{e^{2}} + \frac{143 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} a c^{2}}{e^{4}} + \frac{5 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} - 1638 \,{\left (e x + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (e x + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(15015*sqrt(e*x + d)*a^3 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*
c/e^2 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 +
315*sqrt(e*x + d)*d^4)*a*c^2/e^4 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^
2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c
^3/e^6)/e

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Fricas [A]  time = 1.86052, size = 478, normalized size = 2.39 \begin{align*} \frac{2 \,{\left (1155 \, c^{3} e^{6} x^{6} - 1260 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 18304 \, a c^{2} d^{4} e^{2} + 24024 \, a^{2} c d^{2} e^{4} + 15015 \, a^{3} e^{6} + 35 \,{\left (40 \, c^{3} d^{2} e^{4} + 143 \, a c^{2} e^{6}\right )} x^{4} - 40 \,{\left (40 \, c^{3} d^{3} e^{3} + 143 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} + 2288 \, a c^{2} d^{2} e^{4} + 3003 \, a^{2} c e^{6}\right )} x^{2} - 4 \,{\left (640 \, c^{3} d^{5} e + 2288 \, a c^{2} d^{3} e^{3} + 3003 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^3*e^6*x^6 - 1260*c^3*d*e^5*x^5 + 5120*c^3*d^6 + 18304*a*c^2*d^4*e^2 + 24024*a^2*c*d^2*e^4 + 15
015*a^3*e^6 + 35*(40*c^3*d^2*e^4 + 143*a*c^2*e^6)*x^4 - 40*(40*c^3*d^3*e^3 + 143*a*c^2*d*e^5)*x^3 + 3*(640*c^3
*d^4*e^2 + 2288*a*c^2*d^2*e^4 + 3003*a^2*c*e^6)*x^2 - 4*(640*c^3*d^5*e + 2288*a*c^2*d^3*e^3 + 3003*a^2*c*d*e^5
)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 48.8004, size = 563, normalized size = 2.82 \begin{align*} \begin{cases} - \frac{\frac{2 a^{3} d}{\sqrt{d + e x}} + 2 a^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{6 a^{2} c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{6 a^{2} c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 a c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{6 a c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 c^{3} d \left (\frac{d^{6}}{\sqrt{d + e x}} + 6 d^{5} \sqrt{d + e x} - 5 d^{4} \left (d + e x\right )^{\frac{3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac{5}{2}} - \frac{15 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{2 d \left (d + e x\right )^{\frac{9}{2}}}{3} - \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{6}} + \frac{2 c^{3} \left (- \frac{d^{7}}{\sqrt{d + e x}} - 7 d^{6} \sqrt{d + e x} + 7 d^{5} \left (d + e x\right )^{\frac{3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac{5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac{7}{2}} - \frac{7 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{3} + \frac{7 d \left (d + e x\right )^{\frac{11}{2}}}{11} - \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{6}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{3} x + a^{2} c x^{3} + \frac{3 a c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a**3*d/sqrt(d + e*x) + 2*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 6*a**2*c*d*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*a**2*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(
d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 6*a*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d
+ e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 6*a*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sq
rt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)
/9)/e**4 + 2*c**3*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(
5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 + 2*c**3*(-d**7/sqrt(d
+ e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) -
7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((a**3*x + a
**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7)/sqrt(d), True))

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Giac [A]  time = 1.34009, size = 302, normalized size = 1.51 \begin{align*} \frac{2}{15015} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a^{2} c e^{\left (-2\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} a c^{2} e^{\left (-4\right )} + 5 \,{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{x e + d} d^{6}\right )} c^{3} e^{\left (-6\right )} + 15015 \, \sqrt{x e + d} a^{3}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*e^(-2) + 143*(35*(x*e +
d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*
a*c^2*e^(-4) + 5*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(
7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^3*e^(-6) + 15015*sq
rt(x*e + d)*a^3)*e^(-1)