Optimal. Leaf size=215 \[ -\frac {g^3 (3 d g+4 e f) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}+\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.67, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1635, 641, 217, 203} \[ \frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}-\frac {g^3 (3 d g+4 e f) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 641
Rule 1635
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^4 f^4-12 d e^3 f^3 g-18 d^2 e^2 f^2 g^2-12 d^3 e f g^3-3 d^4 g^4}{e^4}+\frac {5 d g^2 \left (6 e^2 f^2+4 d e f g+d^2 g^2\right ) x}{e^3}+\frac {5 d g^3 (4 e f+d g) x^2}{e^2}+\frac {5 d g^4 x^3}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^4 f^4-12 d e^3 f^3 g+42 d^2 e^2 f^2 g^2+68 d^3 e f g^3+27 d^4 g^4}{e^4}+\frac {30 d^2 g^3 (2 e f+d g) x}{e^3}+\frac {15 d^2 g^4 x^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\frac {15 d^3 g^3 (4 e f+3 d g)}{e^4}+\frac {15 d^3 g^4 x}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {\left (g^3 (4 e f+3 d g)\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^4}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {\left (g^3 (4 e f+3 d g)\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {g^3 (4 e f+3 d g) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.74, size = 168, normalized size = 0.78 \[ \frac {\frac {\sqrt {d^2-e^2 x^2} \left (15 d^3 g^4 (d-e x)^3+2 (d-e x)^2 (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )+3 d^2 (d g+e f)^4+2 d (d-e x) (e f-9 d g) (d g+e f)^3\right )}{d^3 (d-e x)^3}-15 g^3 (3 d g+4 e f) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{15 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.26, size = 624, normalized size = 2.90 \[ -\frac {7 \, d^{3} e^{4} f^{4} - 12 \, d^{4} e^{3} f^{3} g + 12 \, d^{5} e^{2} f^{2} g^{2} + 88 \, d^{6} e f g^{3} + 72 \, d^{7} g^{4} - {\left (7 \, e^{7} f^{4} - 12 \, d e^{6} f^{3} g + 12 \, d^{2} e^{5} f^{2} g^{2} + 88 \, d^{3} e^{4} f g^{3} + 72 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (7 \, d e^{6} f^{4} - 12 \, d^{2} e^{5} f^{3} g + 12 \, d^{3} e^{4} f^{2} g^{2} + 88 \, d^{4} e^{3} f g^{3} + 72 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{5} f^{4} - 12 \, d^{3} e^{4} f^{3} g + 12 \, d^{4} e^{3} f^{2} g^{2} + 88 \, d^{5} e^{2} f g^{3} + 72 \, d^{6} e g^{4}\right )} x + 30 \, {\left (4 \, d^{6} e f g^{3} + 3 \, d^{7} g^{4} - {\left (4 \, d^{3} e^{4} f g^{3} + 3 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (4 \, d^{4} e^{3} f g^{3} + 3 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (4 \, d^{5} e^{2} f g^{3} + 3 \, d^{6} e g^{4}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, d^{3} e^{3} g^{4} x^{3} - 7 \, d^{2} e^{4} f^{4} + 12 \, d^{3} e^{3} f^{3} g - 12 \, d^{4} e^{2} f^{2} g^{2} - 88 \, d^{5} e f g^{3} - 72 \, d^{6} g^{4} - {\left (2 \, e^{6} f^{4} - 12 \, d e^{5} f^{3} g + 42 \, d^{2} e^{4} f^{2} g^{2} + 128 \, d^{3} e^{3} f g^{3} + 117 \, d^{4} e^{2} g^{4}\right )} x^{2} + 3 \, {\left (2 \, d e^{5} f^{4} - 12 \, d^{2} e^{4} f^{3} g + 12 \, d^{3} e^{3} f^{2} g^{2} + 68 \, d^{4} e^{2} f g^{3} + 57 \, d^{5} e g^{4}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{8} x^{3} - 3 \, d^{4} e^{7} x^{2} + 3 \, d^{5} e^{6} x - d^{6} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.40, size = 411, normalized size = 1.91 \[ -{\left (3 \, d g^{4} + 4 \, f g^{3} e\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\relax (d) + \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left ({\left ({\left (15 \, g^{4} x e - \frac {2 \, {\left (36 \, d^{5} g^{4} e^{10} + 64 \, d^{4} f g^{3} e^{11} + 21 \, d^{3} f^{2} g^{2} e^{12} - 6 \, d^{2} f^{3} g e^{13} + d f^{4} e^{14}\right )} e^{\left (-10\right )}}{d^{4}}\right )} x - \frac {45 \, {\left (3 \, d^{6} g^{4} e^{9} + 4 \, d^{5} f g^{3} e^{10} + 2 \, d^{4} f^{2} g^{2} e^{11}\right )} e^{\left (-10\right )}}{d^{4}}\right )} x + \frac {5 \, {\left (21 \, d^{7} g^{4} e^{8} + 28 \, d^{6} f g^{3} e^{9} - 6 \, d^{5} f^{2} g^{2} e^{10} - 12 \, d^{4} f^{3} g e^{11} + d^{3} f^{4} e^{12}\right )} e^{\left (-10\right )}}{d^{4}}\right )} x + \frac {5 \, {\left (36 \, d^{8} g^{4} e^{7} + 44 \, d^{7} f g^{3} e^{8} + 6 \, d^{6} f^{2} g^{2} e^{9} - 12 \, d^{5} f^{3} g e^{10} - d^{4} f^{4} e^{11}\right )} e^{\left (-10\right )}}{d^{4}}\right )} x - \frac {15 \, {\left (3 \, d^{9} g^{4} e^{6} + 4 \, d^{8} f g^{3} e^{7} + d^{5} f^{4} e^{10}\right )} e^{\left (-10\right )}}{d^{4}}\right )} x - \frac {{\left (72 \, d^{10} g^{4} e^{5} + 88 \, d^{9} f g^{3} e^{6} + 12 \, d^{8} f^{2} g^{2} e^{7} - 12 \, d^{7} f^{3} g e^{8} + 7 \, d^{6} f^{4} e^{9}\right )} e^{\left (-10\right )}}{d^{4}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 1030, normalized size = 4.79 \[ -\frac {e \,g^{4} x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 d \,g^{4} x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {4 e f \,g^{3} x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {9 d^{2} g^{4} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {12 d f \,g^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e \,f^{2} g^{2} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {d^{3} g^{4} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {6 d^{2} f \,g^{3} x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {9 d \,f^{2} g^{2} x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {2 e \,f^{3} g \,x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {12 d^{4} g^{4} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {44 d^{3} f \,g^{3} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}-\frac {2 d^{2} f^{2} g^{2} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {4 d \,f^{3} g \,x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e \,f^{4} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{5} g^{4} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {18 d^{4} f \,g^{3} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {21 d^{3} f^{2} g^{2} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {6 d^{2} f^{3} g x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}-\frac {d \,g^{4} x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}+\frac {4 d \,f^{4} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 f \,g^{3} x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}+\frac {24 d^{6} g^{4}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}+\frac {88 d^{5} f \,g^{3}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}+\frac {4 d^{4} f^{2} g^{2}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {4 d^{3} f^{3} g}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {7 d^{2} f^{4}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {d^{3} g^{4} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}+\frac {6 d^{2} f \,g^{3} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}+\frac {7 d \,f^{2} g^{2} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}+\frac {f^{4} x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d}-\frac {2 f^{3} g x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}+\frac {16 d \,g^{4} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4}}-\frac {3 d \,g^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{4}}+\frac {14 f^{2} g^{2} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{2}}-\frac {4 f^{3} g x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e}+\frac {2 f^{4} x}{15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}}+\frac {32 f \,g^{3} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}-\frac {4 f \,g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.04, size = 1178, normalized size = 5.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^4\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________